How did the Greeks use the poetry and fables to teach values?

Answer 1

They embodied moral tales, to illustrate good, bad, consequences and how it happened, to guide listeners along the path of life.

The values they taught were not necessarily the values we value today. Their underlying philosophy was to do nothing which harms yourself, and to push this, the teaching was about not doing anything which would promote self-harm.

Answer 2

The Ancient Greeks utilized poetry and fables (or, more generally, stories) to teach values to their youth and to each other in various ways. One way arrived at religious festival-times, with plays and poetry being offered up as reminders of what it means to be good, or at least to be a good Greek. Another way was more ordinary but perhaps even more powerful: famous stories such as the Homeric epics were very nearly the equivalent of the modern "popular" television shows, as they were recited quite often, in formal and informal situations alike, and as nearly everyone was able to quote from them, and as they cemented society with frequent reminders of basic customs, principles, and values of importance.