Beginning in 1520, right after Martin Luther's Reformation, bloody religious wars began that kept recurring almost yearly for over a century. The Reformation's rapid spread drew battle lines accross Europe as Catholic armies were fielded to crush the Protestants into submission, engulfing the continent in intermittent war from the 1520s to the 1640s. But what was the real cause of these recurring wars? In other words, "Why not live and let live?"
Religion was just the tool, mostly.
The real conflict was over who had the "monopoly on violence" - sovereignty. In medieval Europe, the state formed a useful alliance with the church - the church attested that the king was God's chosen (and thus the only rightful ruler of a given land), and provided a certain level of inter-national balance by keeping the individual fiefdoms, kingdoms, duchies and what have you small (and weak) enough.
But the medieval age was ending as the states gained more power - mostly through trade and craft, which were a lot easier to manage than the huge fiefs. Now you had three strong main powers - the landed, the burgeois and the clergy. And the pope started to feel a lot less important to the rulers who now had a solid power base without relying on their "divine right to rule". The pope didn't like this very much - not because of theological reasons, but as a simple matter of worldly power; if he let the reformation take its place without challenge, he would lose everything. While each of the "reformed" christianities had its own motivations, they pretty much all agreed on one thing - the pope shouldn't dictate what we can or cannot do. Think of how your own countryheads would react if half the people just went "we're not going to pay taxes anymore, thanks".
So on one side, you have people fighting for some sort of freedom, and on the other, people very much happy with the current state of things. The catholic rulers were promised reward in return for their contributions to the wars (as well as continued support in the "divine" department), while the various "reformed" wanted to break that power block apart (some actively, some defensively - my country's Hussites were mostly said to be concerned with keeping the Bohemian crown lands sovereign and letting everyone else do whatever; they got a Crusade for their troubles). And worse, the countries rarely switched "en masse" - so you might have half-catholic, half-reformed people fighting each other, which has historically often resulted in very bloody wars. The catholic church never head a threat of this scope - if they wanted to keep any power at all, they had to do everything they could to hold on. They saw the reformation as a challenge to their rule - which was true, of course, to various extents. Both the worldly and the divine rulers were just humans, despite what the church claimed - and they didn't want to lose any power.
This of course isn't unique to religion. Any time you have someone with violent power, they're going to severely resent anyone challenging that power. The main thing that made the Reformation stand apart was that it was rather sudden (at the timescales usual for change in medieval times) and engulfed pretty much all Europe almost at once, while also being combined with many other "partisan" groups - people were getting a taste for freedom and started challenging their traditions. Why should only the clergy drink the Lord's blood in the church? The Bible said all the people are supposed to eat and drink Lord. Needless to say, literacy had its influence on this - more and more people could suddenly read the Bible for themselves, rather than having it orated by some clergyman; and don't get me wrong, there were clergymen that were quite unhappy with the state of things as well (such as Jan Hus, whose death basically started the Hussite wars).
Other examples of similar bloodshed were in the jacobin revolutions ("democrats") and communist revolutions ("socialists"). It's probably not a surprise that they were mostly civil wars - few safe havens, lots of mistrust even on "friendly ground", lots of fervour. The length of the wars itself was just an artifact of their time, though - even during the 16th century, wars were still rather small, long and drawn out, with few battles that could be considered "decisive", with an almost endless supply of new soldiers and a lot of maneuvering, and still very much seasonal. You avoided fighting in harvest season and cold weather, which only gave you a month of two of warring every year - while the armies were already large enough to require supply trains rather than relying on "foraging" - something already stretched by fighting so much on "home territory", where you didn't want to anger your own people, basically.
Religious war in Europe may be understood in light of the existing order. First of all, the state and the church were vitally connected. This relationship had existed for centuries. Whether Protestant or Catholic, the state exercised its responsibility to protect and promote the religion of its domain.
Second, religious toleration was at first nearly nonexistent. The idea that two or more religions could exist in the same country at the same time was considered neither possible nor proper to the sixteenth-century mind. This explains the conflict between Catholic and Protestants. But it also explains the intolerance among some Protestants, such as when certain Lutheran rulers persecuted Anabaptists.
Finally, the religious wars were fueled by the political ambitions of the kings of Europe, who saw the highly motivated armies under region's banner as tools to advance their political goals. Nowhere is this more evident than in the 1640s when French Catholics fought Spanish Catholics to aid German Lutherans and Dutch Calvinists.
Most of the answers have revolved around the interconnection of Church and State and power politics, which is definitely a main part of the answer.
But the other factor which is very hard for a modern person to get their head around, is that people back then really believed in heaven and hell and the idea that failing to defend their faith against heresies could damn them forever.
How else to explain the number of people who chose to be burned at the stake or worse, rather than becoming apostates in the eyes of other members of their faith and their God? To a modern person, even to many "true believers", an oath taken under duress is not an oath at all. Back then, it was different.
Perhaps the small life expectancy back then was part of this. It was rare for a man to reach "threescore years and ten" even in peacetime. It was common for children to die before reaching adulthood, for wives to die giving birth, for anyone to die at any time of all sorts of ailments and illnesses. If life was short and uncertain, and eternal life the reward of the faithful, faith would be stronger. These days, most people live long enough to experience the downsides of a long life. Has eternity lost its appeal?
I'ts not about religion, it's about power and money
(Disclaimer: First of all, english is not my main languaje but i'm gonna give it a try, now let's go to the point of this answer.)
It's all start with the fall of Constantinople (1453), with the fall of the "horn of gold" sereval stuff trigger, but for our purposes, it also signified the fall of "Haga sophia", the great Cathedral of Eastern Christianity in the hands of the Muslims, the consecuences of this are several and in a variety of planes.
Now in that time knowledgment already means power but not like in today standard, it was a way (specialy in italy and if you where a member of a powerfull family) to show the rival powerhouse than you are, not just a good citizen -because you pay to an artist to make things for you- but also you are clever and richer enough to identify good works and to keep them with your sources, and nobody, specialy (but not limited to) that geographical area want to have clever and rich enemys.
With the fall of Constantinople the Pontificial States under the banner of the Pope Nicholas V, also known as Tommaso Parentucelli, (which was a member of a pro florentian family) saw something than they didn't see coming, the fall of an important place of faith, wich also means a strike in the moral of the Christian Kindoms, Christianity because of that was lose some points (i know, it's seems hard to believe for today standards of religion, but for them this was an important thing) and a new Christian "Main Temple" must be build, Muslims have the Meca in this time and, of course, it wasn't acceptable not have an emblematic temple to all Christianhood, so then it begin, the modifications of the Sn't Peters Chapel now known like the vatican.
Also several lettered Greeks escaping from the Turkish come to Rome, here, finally with ways of get good translations of Greek, and also with the recovery of some text than weren't exactly recovered since the times of the barbarians invations, the renaissanse start (have in consideration than in that times it wasn't easy to copy text, and the other tecnical limitations they have before wonder why).
This is not the only pope implied in this business, he's not gonna be the one who start the building, not the one who finished, as they have a tendency to change with some frecuency in this times, but have in mind than at this time, some time before and several times after all the popes are gonna be part of some of the powerfull familys of the European boot, the duration of this is the same than the duration of the pontifician states (751 - 1870), which was the reason why the powerfull familys in europe are gonna be willing and trying to take power over it when they can.
As you can guess, to change a little poor roman Chapel into what is now is not an easy task, it takes money, time and also political stabilty, it is going to take some time before in 1513 the pope Leo X actualy start the building of the St' Peter's Basilica, and so by that times, they start to raise taxes from abbods and similar stuffs, this ones in time transfer the cost, ask for donations and etc..., by that same time (1513) Germany was both a lot more poor and less educated than italy, so certain guy in charge of an economate (Martin Luther) in 1517 complains about the tax raises and other ways of payment demanded by the vatican, he also see the opulence of Rome and knowing the poor economical status of germany, well, he complain, this -in not so much time- bring the famous 95 thesis to the news at the time also.
But Pico de la mirandola in fact create 900 thesis 30 years before luther, the difference?, well, they aren't "more radical" or something, part of the 95 are contained in the 900 with ease, what really happend is than certain German prince saw this like an oportunity to have more power and remove the power of the King of Germany and spain in that time, no other than Charles V, the Holy Roman Emperor.
You see, opose to a king was opose to the god's chosen-to-be-king in the social imaginary of the Hierarchy and order (Charles Taylor), once removed you can justify easier your conquer if you where a noble, an thus, justify to govern, but it wasn't like that if you fail, if the king is justify in the eyes of him's population and you want to take him out the throne you need to make him dominion weaker, so you can archieve this in three ways:
1) Or "something" take out the king and you are the next in the succession line. (not the case of this future-to-be protestant kings)
2) Or you are a person with enough military power to take over the territorys (not the case of this guys either, not every region of germany are unified with this idea, several of the german princes where actualy pro king, so you have a civil war between your hands and with the plus of the king's Spain army, plus German army attackig you in your own fields, clearly not a possible victory)
3) Or you make him lose legitimacy in order to try to expell him's power, at least of germany, after all, he is Spanish by birth.
This was the one who run, some german princes go in favor of this thesis and use them to burn down the legitimacy of the king and gave themself legitimacy on them own.
This is notoriusly evident when you think than after the reform was already started and start the religius war in germany field, when Luther says "no violence" nobody gives a damn (between the princes) or when the princes, arguing than the divine will rest on them hands make them peasants work without rest, and in sundays, and luther say something about it, nobody -again- gives a damn.
This is also evident when another protestant groups start to appeard in the area, because maybie you don't want to use the justification of your rule in one guy who is saying stuff than you don't want to be a case, so you take some other bishop or some other guy who have enough credibility to do it, and go and support him, that's why several, i mean, a lot of the combats in the religious wars in germany where between different protestant groups.
To be all fair, Luther also send some letters specificaly speaking about anabaphtists, calling them "dogs with sarna" between other names and calling to the destruction of those 'filthy peasants' than opose the rule of them princes, yeah, i'm speaking about that two years period of the Campesine Revolts inside the religious wars (150 years aprox).
In somewhat justice to the plan of the German Princes, the fact than the Spanish empire where that big and the thing than the German prince where playing local can help them to win, in that sence there was a good think, but the guy's simply never put themself together to the start, and that cause a lot of pain, to a lot of people, in fact, nobody can be happy when family fight against family or neiborgh against neiborgh.
Ok, so they tend to fight; we knew that. But why the decrease and then increase in certain specified time range? That’s the question!
Because the time range is chosen that way and the reason limited to religion? There always are all kinds of war, but it is much nicer for your ego to "fight for the right cause" than for power and possession. Think of crusades. Think of saxons. Think of the wars led by karolingian princes. But, these just were wars against pagans, not truly religious people, eh? ;) As for the development necessary to get a decent amount of people with different religious beliefs to form a new church, have a look at clerical history. Pre-Lutherian "dissenters" were called heretics and killed by the church. Took a lot of dissatified people to overcome their fear and speak out loudly. For the rest - see George A. Strieby's answer.
All apes (including humans) fight each other for no reason at all. Humans are different in the sense that they are able to learn advantages of living in peace - it just takes a few hundred years of learning (*).
Moreover, this time there was a reason to fight: Reformation paved the way for development of commerce and industry and it upended existing order of the things in a big way.
(*)I suspect that 'learning' above is just a regular evolutionary process: as opposed to other apes humans improve weapons quickly so ones who love war have better and better chances to die young.
The printing press.
Information control gives social control. Any new widely available information technology gives a surge in social power to everyone but it risks destroying information control. When social control fails conflicts flare up one way or another, until some kind of social order is re-established.
Religion was a code word for the established belief-system or frame of reference. Controlling the frame of reference of peoples mindset is what gives social power and religion was completely crammed with ways to do that back in the days.
If my point would be true then anytime said control would be re-established you would probably be having a real tough time finding any sources of how things were different earlier and the ones you may find would likely not be deemed reputable. Martin Luther was probably not reputable according to the catholic church. Probably a bunch of conspiracy theories.
One way I could try and explain: If you are not especially interested in european history and were born rather recently you would probably not even think of some german monk when you hear the name. It shows that information control can create a shift in narrative which by it's nature is something that limits the thoughts we are able to think.
( Also removed banana. )