Odysseus has lost his way on the voyage home from the Trojan War. He and his hungry crew make a stop in Sicily at Mount Aetna, which is inhabited by Cyclopes. They come upon the Satyrs and their father Silenus, who have been separated from their god Dionysus and enslaved by a Cyclops (named Polyphemus in the Odyssey). These characters are not contained in the Odyssey's version of the event. Their addition provides much of the humor due to their cowardly and drunken behavior.
When Odysseus arrives he meets Silenus and offers to trade wine for food. Being a servant of Dionysus, Silenus cannot resist obtaining the wine despite the fact that the food is not his to trade. The Cyclops soon arrives and Silenus is quick to accuse Odysseus of stealing the food, swearing to many gods and the Satyrs' lives (who are standing right beside him) that he is telling the truth. His son, a younger and more modern Satyr, tries to tell the truth to the Cyclops in an attempt to help Odysseus. After an argument, the Cyclops brings Odysseus and his crew inside his cave and eats some of them. Odysseus manages to sneak out and is stunned by what he has witnessed. He hatches a scheme to get the Cyclops drunk and burn out his eye with a giant poker after he has passed out from inebriation.
The Cyclops and Silenus drink together, with Silenus attempting to hog the wineskin for himself. When the Cyclops is drunk, he says he is seeing gods and begins to call Silenus Ganymede (the beautiful prince Zeus made his immortal cup bearer). The Cyclops then steals Silenus away into his cave, with the implication that he is about do something sexual to him. Odysseus decides to execute the next phase of his plan. The Satyrs initially offer to help, but later chicken out with a variety of absurd excuses when the time for action actually comes. The annoyed Odysseus gets his crew to help instead, and they burn out the Cyclops' eye.
He had told the Cyclops earlier that his name was 'Noman' or 'Nobody' (Greek outis or mētis), so when the Cyclops yells out who was responsible for blinding him, it sounds like he is saying "No man blinded me". In addition to this pun, there is a less easily translated joke based on the fact that the form of "no man" (mētis) is identical to the word for cleverness or art. The Satyrs have some fun with him over it. Odysseus makes the mistake, however, of blurting out his true name as a result of his big ego. Although he successfully makes his escape, the rest of the troubles Odysseus faces on his voyage home are related to this act, as he then faces the wrath of Poseidon, the father of the Cyclops.