By alluding to other texts and stories, Eliot is able to bring in the entire theme, mood, feeling and plot of those other stories, with just one word or phrase. It is almost like providing an internet link right in the middle of a story; you "click" on it, and it brings you to an entire other page of information and applications. The "links" in this poem are several, and add a lot of dimension to Prufrock's plight.
One of the allusions is to Hamlet. He has spent the entire poem wondering if he should "disturb the universe" by asking "some overwhelming question" to a woman; he goes back and forth, being wishy-washy and fearful. He wonders if it "would have been worth it, after all," to have asked the question just to have her respond, "That is not what I meant at all. That is not it, at all." Well, he finally decides, no, it would not be worth it. Here he states, "I am not Prince Hamlet, nor was meant to be," meaning, he was not a royal, brave, conniving, eloquent, profound man (as Hamlet was). Instead, he says that he was "deferential, cautious, a bit obtuse; at time, indeed, almost ridiculous-almost, at times, the Fool." Alluding to Hamlet helps us to understand that he feels enormous pressure; Hamlet is a famous tale of a man sent on an enormous task of revenge-he gives fancy and profound speeches, he greatly impacts everyone around him. Prufrock is perhaps feeling that what he has to say is the same, but unlike Hamlet, he's not up to the task of doing it well.
One last allusion I'll mention is the mermaids that Prufrock hears "singing, each to each," but feels that he does not "think that they will sing" to him. Mermaids are mythical, beautiful creatures that often, in stories, lured seamen to their deaths. Prufrock can hear the alluring tune of the beautiful women in his life, of the potential of being with creatures he considers unattainable as mermaids, because they won't sing to him. He feels unworthy, plain, out of their league, and rejected by the beautiful creatures around him.
Those are just a couple of the allusions in the play, and I hope that helps a bit! Good luck!