It was an oral tradition. Recitation was how it was passed on before it was finally written down. You see the echoes of this in a lot of early Greek writing, they contain a large amount of quotes out of Homer. Being able to navigate this maze of "Darmok and Jalad"-speak with Homer quotes meant you were well educated.
Only thing I've found that comes close is some of Augustine's writing. He does the same with Bible quotes in his Confessions.
I know, I learned ancient greek in school, the history and writing was a large part of that. Most (I'd guess > 95%) of the Greeks weren't able to recite the Illiad if I remember that correctly, oral tradition or not. Even the best educated of them wouldn't really have been able to do so. They were familiar with the content, of course, just like a lot of people in what we consider the West are familiar with what the bible is about.
But I'd be happy to be proven wrong about that, school was a long time ago :)
Only thing I've found that comes close is some of Augustine's writing. He does the same with Bible quotes in his Confessions.