Getting Polyphemus drunk and asleep was part of their plan to escape. Once he falls asleep, they'll heat up their weapon.
"'Cyclops try some wine. Here's liquor to wash down your scraps of men. (Homer Line 256)'"
Polyphemus asks for Odysseus' name. Odysseus tricks him by saying his name is "Nohbdy." If Polyphemus rages about Odysseus, he'll say that nobody did anything to him .
"'... My name is Nohbdy; mother, father, and friends, everyone calls me Nohbdy.' (Homer Lines 274-275)"
Odysseus and his men are heating up their weapon to blind Polyphemus.
"The pike of olive, green though it had been, reddened and glowed as if about to catch. (Homer Lines 286-287)"
"I drew it from the coals and my four fellows gave me a hand, lugging it near the Cyclops as more than natural force nerved them; straight forward they sprinted, lifted it, and rammed it deep in his crater eye, and I lean on it turning it as a shipwright turns a drill in planking, having men below to swing the two-handed strap that spins it in the groove. (Homer Lines 288-295)"
Odysseus and his men stakes Polyphemus in the eye.
"'Nohbdy, Nohbdy's tricked me, Nohbdy's ruined me!' (Homer Line 317)"
Since Odysseus fooled Polyphemus by saying his name was "Nohbdy", Polyphemus' siblings believes that nobody tricked or harmed him. Polyphemus' siblings doesn't help him.
In the morning, he'll release his sheep outside which also means that Odysseus and his men too because they are under them. To make sure the sheep are the ones leaving, Polyphemus strokes them.
"Three abreast I tied them slightly together, twining cords of willow from the ogre's bed; then slung a man under each middle one to ride there safely, shielding left and right. (Homer Lines 337-340)"