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I think that first we need to make some clarity: in Ancient Greece, it was not science itself in its modern sense that appeared, but philosophy.
What is the reason for the development of philosophy in the Athenian state?
I think there are two reasons for this:
A democratic system of governance in Athens. The most important decisions were made at the national assembly, and oratory was of great importance – it was necessary to convince the members of the Assembly that they were right. Thus, it was necessary to be able to find arguments for successfully solving this problem.
The presence of a large number of residents who are not included in the government system, but serve all the needs of the city. we are talking about slaves and meteks. Free (and well-off) citizens of Athens could afford to do things that were quite abstract and not aimed at ensuring their survival. Everything, as far as I know, came down to the development of the mind and body.
Despite the fact that mainland Greece was the central region of the Hellenistic world and the birthplace of three of the greatest philosophers, philosophy penetrated there last. Ancient Greek philosophy spread from Asia Minor (Thales, Anaximander, Anaximenes, Heraclitus, etc.) and Italy (the Pythagoreans).
There are various hypotheses about the origin of philosophy and science, which cannot be confirmed or refuted. Any culture at a certain stage of development moves from a mythological picture of the world to attempts at a more rational understanding. I think an important role was played by the huge diversity of ancient Greek civilization (many states with very different cultures and political structures), the relatively high standard of living of the free population and developed trade.