Before all else, I want to point out that this is a sensible topic and it is easy to go off the well-meaning path — my motives are strictly curiosity and I believe this site can actually have a fruitful discussion around it. I would also like to mention that I have a quite limited, and very Europe/West-centric history knowledge, so please acknowledge my bias/ignorance. I don’t even know enough about the Western part, let alone enough to compare it with other cultures’ respective systems at the time.

I have talked about this topic numerous times with friends and while we had some theories why it might have started there, none of them were particularly convincing. Nonetheless, here are a few reasons we could come up with (of course it is multi-faceted), please expand/argue about them if you feel, or write new ones as well (I try to be brief here, partially due to hitting the edges of my knowledge in many cases, but also to not bias people to any particular topic):

Religion, Philosophy

Christianity is/was the predominant religion of the region. Plenty of important discoveries/inventions came from monks “learning about God through the natural world”, and many branches of Christianity believed in sharing not only their religion, but knowledge as well - Jesuits being particularly famous for teaching, and collecting vast amounts of knowledge. (Obviously, the Catholic Church had plenty of wrongdoings as well, and was not opposed to keeping people in the dark)

Another notable religion which deserves a mention in this topic is Jewish. I am unfortunately even less knowledgeable about it, but the number of Jewish scientists and discoveries is staggeringly high. From what I gathered, they have a strong culture of arguments/discussions regarding the Torah, and blind acceptance is not the way (as opposed to the Bible), which might have some relevance.

The Arab world had an important relationship with Europe, but from this perspective it may not be as important.

With that said, I really can’t argue whether these are significantly different than, say, Eastern religions. Maybe the Abrahamic three are a bit more individualistic, which might have a bigger relevance here in the direction Western philosophy has gone.

Economic system

I know even less about this topic. One important point I do know a bit about is the question of loans, which were forbidden by the Christian Church (for a while) and Islam, but not for Jews (so loans could be obtained in Europe). Not sure if there is a difference between different parts of the world here that is of relevance.

Capitalism itself is also a result of these philosophical ideas, so there might be more to this financial aspect.

Society, societal norms

A prominent theory might be the existence of universities. While different kinds of school systems have existed earlier - to my ignorant knowledge - Europe was ahead in terms of education. This is quite self-explanatory, accumulating smart people and ideas together greatly improves their “productivity”.

Colonialism

Self-explanatory, but if this is the answer, I would also like to extend the question to why the Western world was the first at that at such an extent? That also required technological innovation and a motivation for it. Though it itself has plenty possible explanations like good geographic location.

Climate, geography

Europe has a generally mild climate, well-fit for efficient agriculture and animal husbandry. It also has numerous rivers. It is not the Biblical garden where you have fruits all year long, so you do have to rely on your own work to survive winters, but it is definitely not the desert neither. But that is not a convincing answer either, because plenty of regions are similar, and early scientific advancements come from the more desert-y Arab word, with perhaps harsher conditions.
Europe is also well-connected to other regions.

War, Politics

I quite liked this answer one of my friends came up with (within our circles that is) — due to many, small political entities being so close to each other, there were lots of conflicts, many of which resulted in wars — which are significant drivers of “innovation” even today. Pardon my ignorance, but other regions were mostly ruled by huge empires, that later fall apart and were conquered again, or small settlements with little outside contact. This necessitated less novelty in those conquests.

But even non-war conflicts themselves could have had positive effects, the aristocracy of Europe had strong connections with shared events - besides the more “romantic”/formal aspects, it was also a hub for many intellectuals from different countries to engage in deep discussions. Add to it that most research/discovery/philosophy came initially from people of aristocratic backgrounds.

I could write many more, but I am afraid that would have even less substance. I would be very interested in your input on this, what unique context allowed this intellectual growth that resulted in many aspects of our modern lives we take for granted?