What happens after Odysseus and his men become trapped in the Cyclops's cave?
Polyphemus, the Cyclops, sets the huge stone door in place when he returns from shepherding his flock. It is a stone so large that Odysseus and his men could never hope to move it. They beg for the hospitality that Greek culture has enabled them to expect, as Zeus was the god of travelers, and it was believed that offering hospitality was, thus, a religious imperative. However, Polyphemus says, "'The Cyclops pay no heed to aegis-bearing Zeus, nor to the blessed gods; because we are much stronger than themselves.'" This is bad news for our hero because the Cyclops can clearly overpower them. And he does.
Polyphemus immediately kills and eats two of Odysseus's men, and then goes to sleep. Odysseus realizes that they cannot simply kill this monster or else they will not be able to move the stone from the door, and they, too, would perish inside. When the Cyclops awakens in the morning, he eats two more men, and two more again that night.
However, after dinner, Odysseus gets him very drunk on the undiluted wine he has brought with him. (Wine used to be very viscous and would be mixed with water; one could make their wine as strong or weak as one desired. Odysseus plies Polyphemus with the full-strength stuff, and not even the giant monster can remain sober.) Odysseus tells him that his name is "No man" or "Nobody" (depending on your translation). When the Cyclops passes out drunk, Odysseus and his men sharpen a large olive stake in the fire and plunge it into the monster's one eye, blinding him. Polyphemus screams, and when his fellow Cyclopes come to check on him, he tells them that "No man" is hurting him, and so they leave without offering him any aid! (This is just one example of Odysseus's cunning.)
Finally, when the Cyclops rolls aside the stone at the door to let his flock out, Odysseus and his men tie themselves on the underside of the sheep (three sheep across hold one man) so that Polyphemus will not feel them on the sheep's backs as they run out the door. This is how they make their escape.