When Homer composed his epics, Greece was ending a dark age that started 300-400 years earlier with the fall of Mycenae (the Bronze Age collapse, the Sea Peoples, etc).
Writing was forgotten and oral tradition reigned supreme.
He learned the stories himself and created his own narratives. In a sense, he would have been more like Shakespeare, who did essentially the same exact thing. (Many of his fictional plays are based on stories by Ovid.)
As with later works, such as De Bello Gallico by Julius Caesar, Homer's works would have been read out loud, written copies being too expensive for any but the wealthy.
They were usually read one chapter at a time, not the whole work.
They wouldn't have been performed on stage, just read out loud.
I'd probably compare it to radio shows in the early 20th century. People would sit around the radio and all listen to the latest adventures of The Shadow or Superman.
If you weren't at the agora in your Greek town or not by the radio at the right time, you'd have to ask someone to fill you in on what you missed.