Let's agree for sake of argument that all cultures have equally impressive bodies of oral tradition that include stories about the origins of things and about the conduct of powerful supernatural beings and heroes. I think that's a fair statement, but it is difficult to evaluate a subjective term such as "equally impressive" - but we are agreeing on that premise to answer your question.

First of all, the Greeks were good at codifying their stories in writing. They weren't the only culture that did this, but many did not, so those that did not go through this process could participate in the international contest of being recognized and celebrated in the same way.

So then the question focuses on why the Greek tradition is celebrated more than others that were recorded (the Egyptians come to mind, for example). While the Romans conquered both Greece and Egypt, they tended to recognize kindred supernatural beings and stories in the Greek tradition. And given what seemed to them to be the eloquence of Greek authors and poets, their material tended to be promoted as a requirement to achieve cultural literacy for an educated Roman. In turn, subsequent European society tended to continue this notion. The fact is, even today, it is difficult to read and understand the greater body of western literature or to view much of western art without understanding the many references to ancient Greek stories. The only other body of ancient stories that occupies a similar role when it comes to cultural literary would be ancient Hebrew traditions as manifested in the Old Testament.

Stories of other cultures can be just as remarkable and enlightening, but none have been as thoroughly and consistently integrated into western literature and art than these two traditions. One can contest why it is necessary to understand western art and literature for a sense of modern cultural literacy - but that is a separate discussion. As long as understanding western art and literature is essential, understanding Greek mythology (and its ancient Hebrew counterpart) will be required.

There's already a great answer in here, but there's also the fact that Greek civilization has never really "collapsed". No matter how many times Greece was conquered, it was never replaced by another culture. And, as other posters have pointed out, Greek was the language of learning for a long time. Tl;dr: Greek culture has never been dominated by another culture