The Greeks had a very sophisticated and innovative government structure: democracy. In addition, they had a thriving culture in terms of art and drama. It seems to me that people were able to express themselves well in Ancient Greece, and this led to their success.
I thought it might be helpful to you to provide some bibliographical leads by doing some internet searches. The links below may be useful to you.
Clearly the contribution that Greek culture made in the area of the arts is of inestimable value. So much of the development of tragedy and drama in particular stems from Greek drama, and the concept of tragic heroes as people that have tragic flaws is still a much-debated idea in the study of more modern tragedies. Aristotle's definition of a tragedy forms an interesting lens which we can use to critically interrogate dramas today.
I respectfully disagree with post number 2 that the advanced in Greek thought were the result of the Macedonian rise; in fact the rise of Macedonia only occurred after the Greek city-states had virtually destroyed each other in the Pelleponnesian Wars. Prior to that time, Macedonia had been a minor nuisance.
The Greeks had traded early on throughout the Mediterranean basin for several hundred years. The end result was a great expansion in Greek learning and thought. They learned geometry, medicine, and some ideas on religion from the Egyptians. They adopted the Phoenician alphabet, and also blended the learning that gained from other areas with their own intellectual systems, which resulted in a philosophy rooted in human reason. This borrowed learning also extended to art, literature and concepts of morality.
The greatest of the Greek Philosophers was Socrates, who considered honor more important than fame, and that one should reflect on ones own personal goals in life. His ideas were alien to orthodox Greek thinking, and he was sentenced to death as a result. His disciple, Plato, committed his thoughts to writing which Socrates had not done. Plato's work was a relentless search for virtue which he believed could only be attained through searching through the realm of Forms and Ideas. His ideas were quite abstract, and were abandoned by his own disciple, Aristotle, who believed that the mysteries of the world could be deciphered through the rigid application of logic. It was Aristotle who was retained by Philip of Macedon to tutor the young Alexander. It was also Aristotle who was primarily responsible for Greek advances in the sciences, even though many of his ideas (such as a geocentric universe) were incorrect. Other Greek scientific work had been towards developing superior weapons; time and energy saving devices were not needed, since all such work was done by slaves.
The greatest accomplishments of the Greeks to the Arts was the development of Drama in the theater, during which actors wore masks, and performed normally with a chorus. Although many Greek plays have been lost, the works of Euripides, Sophocles, and Aeschylus are still extant. All Greek plays dealt with the relationship of humans to the gods, and were based heavily on Greek religious thought.
The general answer is that the advances of the Hellenistic world were caused by upheaval. The old state of affairs, where Greek poleis dominated the area, had been disrupted and this led to a great deal of questioning.
One of the effects of all this questioning was the rise of new philosophies. During the Hellenistic period, such new philosophies as Epicureanism and Stoicism and Cynicism arose. These were ideas that focused on the issue of how individual people should live, not on issues of what the world was like. I don't now if these would be called "advances," but they were certainly changes that are attributed to the social upheaval caused by the rise of Macedonia.