Who wrote fables in ancient Greece?
This is an ambiguous question, since you do not define what you mean by fables. If you are speaking of the great epics, such as the Iliad and Odyssey, then the best theory is that it was first oral poetry that has been passed down from generation to generation until someone by the name of Homer put it down in writing. As for the the other stories such as the Oedipus trilogy or Medea usually people in Athens wrote them. In particular, Sophocles wrote Oedipus. Other famous writers were Euripides, Aeschylus, Aristophanes.
Though very little is known of him, one of the greatest fable tellers was a man by the name of Aesop. What we know of him has come from first Aristotle, then Herodotus, then Plutarch. He is generally believed to have been born a slave in the Thrace area, around 620 BCE and later set free, since there is a recounting of a story that he had argued a case of some sort for someone; a slave could not have done this. He is considered to have accounted for over 600 fables from his presumed master, some sort of master story teller himself. Aesop died around 560 BCE. How many fables he is responsible for is unknown, but several scholars believe it is approximately 300-500.