Why was Europe the continent to rise to world power?
This is an extremely controversial question and no one knows the answer for sure.
In past times, almost everyone (at least everyone who was white) believed that Europe came to world power because white people were superior. This was an avowedly racist point of view. It held that Europeans were innately better than everyone else and were therefore able to dominate.
Today, that sort of idea is no longer acceptable to most people. However, many people continue to argue that European culture was in some way superior to other cultures. They say that European individualism, for example, created an atmosphere in which European people were more able to invent new technologies and to imagine new ways of doing things. This argument holds that European values were more conducive to things like technological progress, which is the main factor that allowed Europeans to dominate the world. I suspect that the majority of laypeople would hold beliefs similar to these. These views are still held by some academics, as shown in the book that is reviewed in the link below.
The final idea that I will present here is the idea that geographical luck allowed Europe to become dominant. This thesis is presented most famously by Jared Diamond in his book Guns, Germs, and Steel. He says that Eurasia became dominant because of two main factors. First, it was able to get agriculture first because it was home to many domesticable plant and animal species—more than any other land mass. Second, agriculture was able to spread around the land mass because it has a long east-west axis (plants and animals from one latitude can generally thrive in most places to the east and west that are at that same latitude). What this meant is that agriculture arose first in Eurasia and then spread throughout that land mass. This let Eurasia develop civilization first. With civilization comes technology. Since Eurasia had civilization before any other region did, it had more time to develop technology that would allow it to dominate the world.
This does not, of course, tell us why Europe came to dominate the world instead of, for example, China. That question is harder to answer. It is, in my view, more plausible that culture would matter in this situation. If a number of areas of Eurasia had similar advantages in terms of being able to acquire and support agriculture from an early time, it stands to reason that some other factor such as culture might help to determine which area becomes “best” at creating technology. Diamond also argues that geography plays some role here, though he admits he is being more speculative on this question. He says that the irregular coastline and relatively mountainous terrain of Western Europe allowed many small countries to arise while China’s geography made it possible for one government to dominate a huge area of East Asia. This allowed more competition between countries in Western Europe than in East Asia, leading to greater technological innovation.
Again, this is a question that cannot be answered with scientific certainty. The reasons I have laid out here are (or have been) typically seen as some of the most likely reasons for European dominance.