Lincoln's election precipitated the secession, but didn't they have enough congressional power to continue things much as they were? It seems in the preceding two decades the slave states had managed well enough extracting concessions from the free states such as the Missouri compromise and the fugitive slave act. Wouldn't it have made more sense to continue as part of the U.S. and ignore the abolitionist sentiment of many in the North?
The fundamental cause of southern secession (and ultimately the Civil War) was the US's inability to solve slavery at the national level.
The Civil War was not fundamentally about "states rights". Asserting a state's right to secede doesn't speak to why the state wants to secede. Steven's citation of reasons in his answer only serve to underline this. When the northern states were threatened by the War of 1812, they considered secession. When South Carolina was threatened by a tariff, they attempted to nullify the law. When a state's self interests come into play, they'll take advantage of whatever political mechanism they can imagine to assert that self interest, up to and including nullification, secession, and war.
The real story is not in the political mechanics but the underlying interest in preserving slavery that forced the South to become so hell bent on their "states rights". If Northern states had seceded over the War of 1812, we wouldn't assert the fundamental cause was a debate over state's rights. Rather we'd say it was their opposition to the War of 1812. The same applies for the South's secession as well. Their interest in preserving slavery drove them to use untested constitutional mechanism and eventually go to war.
And when you consider the South's conundrum it becomes clearer.
White southerners lived in constant, real fear of slavery/black insurrection. They had experienced violence from slaves during Turner's Rebellion. They looked south to the Haitian Revolution and similar rebellions in the sugar colonies and saw little comfort in how those societies were transformed. Maintaining the institutions associated with slavery was a matter of life/death for white southerners. Laws were passed to further and further restrict the activities of slaves and freed blacks and codify racial distinctions. Even if the injustice of the system might be acknowledged by some southerners, the fear that slaves or freed blacks might (perhaps justifiably) seek vengeance was deep.
Even if a southerner thought that ending slavery was the right thing to do (and many did), the way to get there without providing massive disruption to southern society (economic or otherwise) was very hard to see. Aside from what they would have perceived as lost property, you have the question of what do you do with the slaves? If they're freed won't they just take up arms and overthrow southern society? Would the freed blacks compete with poor whites and drag down wages? In every sense, security, agrarian economics, social interactions, slavery and the associated racial class system was at the heart of Southern life and it was apocalyptic to imagine life without it. Lincoln himself points out this fear during his inaugural speech when trying to be conciliatory
Indeed clauses to maintain of the institution of slavery are rife throughout the Confederate constitution:
Article I, Sec 9:
Article IV, Sec 2 (1):
Article IV, Sec 2 (3):
Article IV, Sec 3 (3)
This need for security and tight adherence to the institution existed in direct contrast to Northern abolitionist's deep, often religiously founded, moral opposition to slavery from people such as John Brown and others. Many movements sprung up to curtail the expansion of slavery such as the Free Soilers and Republicans.
The national politics prior to the civil war became dominated by the South's attempt to keep slavery going and expanding vs the North's determination to keep it from expanding. Southerners understood that the only way to maintain their power in Congress (and ultimately their sense of security) and to grow economically was to fight for the expansion of slavery. The South would uses the power of the Federal government to protect and maintain slavery. The Fugitive Slave Act along with the Dred Scott decision show that the South is happy to use/laud Federal institutions if it helped them meet their end goal. In fact many feared that these were overreaches of Southern influence into Federal power. To quote Lincoln's House Divided speech:
Many in the North resented the power that the 3/5ths compromise, the Dred Scott decision, the Fugitive Slave Act, the annexation of Texas, and the Mexican-American War gave to Southern states and argued that the South had an inordinate amount of power. Indeed Lincoln above is close to asserting a "states right" to deny slavery. In addition some Abolitionists themselves considered secession due to their perception that the South had a large amount of power.
Both the sectional conflicts around expanding slavery and the issue of slavery were ultimately irreconcilable. While campaigning Lincoln attacked the South's hold on Federal power through the continued expansion of slavery through warfare, annexation, law and the 3/5ths compromise. After Lincoln was elected, there was already a shooting war in Kansas and John Brown had raided Harpers Ferry in an attempt to incite a massive slave revolt. Lincoln's party was seen by the South as radically anti slavery, and indeed opposition to slavery was the issue that their party was founded on, to quote this history of the Republican party:
The South's siege mentality would only reach a fever pitch upon Republican Lincoln's election. With what they perceived as a virulently anti slavery, pro Northern party increasing its influence coupled with and ever increasingly violent political far-left political opposition, the south felt deeply threatened politically, economically, and physically. Virginia's secession ordinance while brief, discusses the "oppression of the Southern slaveholding states"
Also as this blog summarizes Civil War historian James McPherson:
They tried whatever mechanism they had, citing state's rights and eventually finding themselves in a war. Without doing so, they feared, they would have change forced on them through incited slave rebellion or what they perceived as revolutionary Republicans. Even if the end of slavery wasn't imminent, the loss of the political power that had been protecting the institution felt deeply threatening due to the magnitude of how fundamental slavery was to southern life. So threatening that they were willing to use whatever constitutional mechanism necessary to preserve their political independence and keep slavery going.
Edit in response to Steven's Edits
The issue isn't whether legally the South had anything to fear at the legal level. The issue was that they had managed to use Federal power during the antebellum period to allow the continued spread and protection of slavery. Despite deep fissures, they had managed to continue to dominate and grow political power to protect their slave holding society. The incoming Republicans threatened that as wikipedia puts it
Its understandable Lincoln would be fairly conciliatory in his inaugural speech. The south was taking action to secede. He wanted to maintain the Union and preferred a peaceful solution. He also has to worry about the slave holding states that remained in the Union. That doesn't change the platform of his political party, the reason the South found him so threatening politically. Lincoln himself acknowledges how apprehensive the South is to the Republican takeover:
He echoes my assertion of how much upheaval white southerners saw in ending slavery.
Regardless of any political or legal institutions, when 1/3 or more of your population is enslaved, you might worry more about the plans of those enslaved. When someone like John Brown comes knocking at your door to incite a revolution, you do what you have to to baton down the hatches, especially when it seems his incendiary, potentially threatening actions appear to be hailed in many in the North. No law is going to prevent the enslaved from taking back their freedom when given the opportunity.
Lincoln was elected from Northern Votes alone. Combined with Northern monopolies in the other branches of government, this was the warning bell to the South that they no longer had any say in politics at all.
The North was industrial, and the South was agricultural. This meant that the South's exports earned more (though obviously this also meant that the North was more prepared to fight a war). The North didn't like earning less money than the South, so they taxed the South. Anytime a Southerner wanted to export their goods, they had a pay a large tariff.
The Southerners felt they were being treated like colonies with no real say in anything. So they did what they did last time they were in that situation, they seceded.
If you're interested in further reading, Facts the Historians Leave Out is a very good book on the subject.
I highly recommend Charles Dew's Apostles of Disunion which lets the words of the secession commisioners speak for themselves.
The most common cause those comissioners themselves state is slavery.
The issues that ultimately led to the secession of the southern states had been brewing for a considerable amount of time. While most people (especially northerners) claimed the Civil War was all about ending slavery, the reasons that the south seceded actually had very little to do with that issue. In fact, the South had won a couple of major decisions that supported slavery, such as the Dred Scott decision in 1857 and the Fugitive Slave Law in 1850.
Perhaps the biggest issue that truly led to the secession was the debate over States' Rights. Going as far back as 1798, there had been an ongoing debate over whether or not the Federal government had the right to pass laws that contradicted laws already in place at the State level. While many people claim that the South was attempting to break up the Union, it was more accurately a case of them attempting to stand up for themselves and declare that the laws passed within a State had superiority. In fact, the preamble to the Confederate States Constitution starts with "We, the people of the Confederate States, each State acting in its sovereign and independent character..."
In 1828 Congress passed tariffs that benefited trade in the northern states. These tariffs ended up being detrimental to the Southern states that relied heavily on the ability to export their agricultural goods. In 1832, the state of South Carolina passed a state law that declared these tariffs to be invalid within the state of South Carolina. Persident Andrew Jackson reacted by sending a naval flotilla as well as the threat of federal troops to enforce the tariffs in South Carolina. This was one of the first major steps in breaking down the relationships between the North and the South. Over the next couple of decades, the South continued to suffer financially while the North benefitted, and eventually the Southern states decided they had put up with it long enough. This is why even now most Southerners refer to the American Civil War as the "War Between the States".
Interestingly enough, while it was the South that actually seceded, it was the northern states who had first threatened it. The New England states were opposed to the War of 1812, and at the Hartford Convention had actually discussed seceding themselves. Even then it was more about doing what was best for the individual states!
EDIT: In response to Doug T's impassioned argument that the war was about slavery.
It is a mistake to assume that Lincoln was "virulently anti-slavery". In fact, in his inaugural address he tried to reassure Southerners by declaring that he had no desire to interfere with slavery where it already existed. Furthermore, prior to his election, Lincoln took a very similar position when he ran for the US Senate. In his acceptance speech he stated "I believe this government cannot endure, permanently, half slave and half free." He then went on to argue that "slavery in the United States would eventually have to end everywhere or become legal everywhere in order for the nation to survive" (emphasis added). While Lincoln as an individual did not condone or approve of slavery, he also did not take an open stance against slavery.
In all fairness, I will concede that slavery was indeed a contributing factor as a reason for the Southern states seceding, and in fairness, I never meant to imply that it was not. While it was true that the institution of slavery had been supported by the Supreme Court, public opinion regarding slavery, particularly in the North, was turning against it. It was becoming increasingly apparent that the ability to expand slavery into new states as they joined the Union was going to be difficult, if not impossible. As a result, they would eventually find themselves isolated away in a country that was growing under the influence of a government that did not support their beliefs.
Once Lincoln was elected, the South decided that their voice was not going to be heard in American politics, so they decided to form their own government based on the principles that they felt benefited them the most. This meant they would be free from unfair tariffs, would have the right to decide on laws that were deemed appropriate for each state, and would permit them to continue to own their slaves as well as possibly expanding the institution by encouraging new states that were developed on the continent to join the Confederacy instead of the Union.