Barring a few places in the erstwhile USSR, for which communism and its fall had a major role to play, countries with a predominantly white population, collectively called "The West" are the most economically (and socially?) well to do ones in the modern world.
Going back a few centuries, it was the Europeans that populated the "new world". Why not people of other regions? Because Europeans had the most resources to do so? But some medieval Asian countries were rich too. Asian, African and South american countries were the ruled upon, not the rulers. Why were the Europeans the colonists, not these other countries? Even if they weren't, why couldn't they keep the invaders out effectively? These countries actually had a very rich and ancient history, with access to considerable engineering and tech for those times. Trade flourished too, think about the silk route and the extensive reach of the Indian subcontinent. But they still did not/do not thrive as much as the Europeans.
I know race had no part to play in this, but why the coincidence? Was it the climate? the geography? The religion? Invention of the steam engine and the following industrial boom? (But colonisation started before that era)
This is the question of modern world history. In fact it is a huge set of questions on which a lot has been written. The Wikipedia article on Great Divergence gives a pretty good summary of some important work on the topic. I won't try to cover everything in there, but I will elaborate on a couple of key points that come to mind based on the original question.
First, we should keep in mind that Europe c. 1450 was not an especially advanced area at all. With the Dark Ages were coming to an end, Europe was perhaps a flourishing civilization, but not the only one. To its east, the Ottoman Empire was at the height of its power and expanding in Europe's direction. (The relevance of the Ottoman-European rivalry to our question has been getting more attention thanks to a recent book, How The West Came To Rule.) It was basically out of desperation to get around the Ottomans that Portuguese sailors began to make some breakthroughs in navigation. This gave them increased maritime contact with Africa, but despite their relatively advanced weaponry and navigation skills, the Portugese were not immediately able to dominate or conquer most of the peoples they found there.
Second, as the question already begins to address, Europe's ascent in the following centuries would not have been possible without the colonization of the Americas. Arguably the most important consequence of European conquest of the Americas was the massive influx of silver from South American mines under Spanish control. This was a central factor in the price revolution that shaped Europe's commercial development, and also in the course of relations between Europe and China. Among the other key reasons that European contact with the Americas mattered so much, aside from the silver, was the triangular trade that came to encompass the Atlantic.
Without being bogged down by multiple stipulations in history I'll attempt to answer this as simply as possible. Most likely Europe advanced rapidly due to mercantilism and the free flow of ideas into Europe due to closer relationships with other cultures from trading. We borrowed the great ideas and inventions of the world then made them our own or developed them much further.
This question is discussed –though not succinctly- in Jacques Barzun’s From Dawn to Decadence. It largely started with an idealized revival of Greek philosophy. But it wasn’t an event but rather a multistep process.
The course to study is western Civilization, though I've heard that it's gone extinct.
Let's take the Non-Western countries of the Modern Age. Last I checked, there were some major imperial Powers who governed various parts of "The East" during this time; The Ottoman Empire, The Russian Tsarist/Romanov Empire, The Persian Savafid-(sp?) Empire, The Moghul Empire of the Indian subcontinent and the Japanese Empire of the Far East. If you examine the historical geography of Asia during the Modern era-(especially the Early Modern era), you will see that a sizable part of the continent was "dominated" by one of these empires for many centuries.
As for the West, the major imperial Powers during much of the Modern era, were Spain, Portugal, the Netherlands, Austria-Hungary, France and Great Britain...(one could perhaps include the trading colonial powers of Venice and Genoa, Italy during this time as well).
So if you look at the historical geography of the Modern era, in particular, the Early Modern era, you will see that much of the world, was governed by imperial powers.
I think you would like to know as to why the Modern European West-(i.e. Great Britain, Northern Italy and the pre-unified Germanic lands), were culturally and intellectually "dominant" during this time period, when contrasted with their Eastern imperial counterparts and perhaps other European countries.
There is really no single answer to this question. As someone mentioned in the Comments section, Jared Dimond's "Guns, Germs and Steel: The Fates of Human Societies" asks this very question and Professor Dimond's answer has much to do with the nature of physical and human Geography, and its relationship to World History. The Conservative Writer Thomas Sowell wrote a number of books and essays on this issue as well.
An answer MAY date back to the Northern Italian Renaissance which began around 1400. With the rise of the Northern Italian Renaissance, cities, such as Florence and Venice, were also very important trade cities with wealthy Patrons, such as the Medici family. The commercial, as well as the cultural wealth that distinguished The Northern Italian Renaissance from many of its contemporaries worldwide, may have also had a distinct geographic advantage. Remember that Northern Italy, is the historic (and contemporary) gateway to continental Europe-(or Northern and Western Europe). Northern Italy had fine Universities dating back to the Middle Ages, such as The University of Padua, the University of Pisa, as well as The University of Bologna. These Universities, along with other Northern Italian cultural institutions, revitalized the ideas and innovations of the Ancient West and incorporated such ideas and innovations into their own narrative and creativity. (The Polish Astronomer Nicholas Copernicus, taught at one of the above mentioned Northern Italian Universities and Galileo taught at The University of Padua).
But the Northern Italian Renaissance was not confined to Italy proper. Great ideas spread from this region into neighboring European regions. And while it is certainly true that the Modern European West produced brilliant minds, such as Locke, Voltaire, Rousseau, Newton, Hegel, as well as many others, each of them FOLLOWED the brilliant innovators of the Northern Italian Renaissance.
One could also look to Early Modern Germany as the birthplace of Modern Western ideas and inventions. Although the country of Germany proper would not come into being until 1870, various German speaking lands did have some cultural all-stars. Johannes Guttenberg of Mainz, Germany and his famed Printing Press which revolutionized Printing and Publishing within the European continent during the Early Modern Age. In the German town of Wittenberg, was the Theological Reformer Martin Luther and his Protestant Reformation, which may not have succeeded had it not been for Mr. Guttenberg's transformative invention. Both Guttenberg and Luther lived during the time of the Northern Italian Renaissance.
And of course there is the commercial explanation as to why the Modern West would "dominate" much of the world. The European imperial powers of Spain, Portugal, England, France and the Netherlands, were ALL...maritime powers. Portugal's, as well as France's backyard (or front yard), is the Atlantic Ocean. Spain has the Mediterranean to its East, the Straits of Gibraltar to its South-(pre-1800's) and the Atlantic to its West and North. These countries had significant geographical "head starts" when compared with other countries around the world, who were either landlocked or had very limited access to major waterways. The domination of global trade, via the oceans and seas, often provided maritime powers-(such as the above mentioned countries), with access to great wealth and power.
If you put all this together, you MAY be able to see as to why the Modern European West was (and still remains), the dominant power in the world for over 600 years. Again, there is no single or universally encompassing or catalyzing explanation as to why this was (and is still) the case; however it MAY offer AN explanation as to why this type of historical reality emerged over the past six centuries.