John the Savage's knowledge and love of Shakespeare help situate him as a representative of an older world order that has been lost in the "brave new world" of the World State. From his quotations of Shakespeare, we learn that what he values and cherishes is a vision of society that directly clashes with the realities of the World State.
For example, John has a traditional notion of romantic love. He falls in love with Lenina, but to him this means he will be faithful to her and she to him: they will be devoted only to each other and eventually marry and start a family. He also has traditional notions that a woman should be sexually chaste before marriage. All of this is completely contrary to how Lenina has been conditioned. To her, it is neurotic and wrong to be devoted exclusively to one person. She believes she should give herself sexually to multiple partners because everyone belongs to everyone else. She has been trained that marriage and childbearing are shameful and disgusting. Thus, when Lenina unzips and steps out of her unitard, and John, in horror, quotes Othello, calling her an "impudent strumpet," we learn his attitude toward the sexual mores of the World State. He goes on in the same scene to quote King Lear on woman's unfaithfulness, saying:
Down from the waist they are Centaurs, though women all above. But to the girdle do the gods inherit. Beneath is all the fiend’s.
In other words, through Shakespeare, John communicates his deep aversion to women having casual sex with multiple partners—a point of view that is comically incomprehensible to Lenina.
Likewise, John's quoting of the phrase "brave new world" from Shakespeare's The Tempest also characterizes him. At first sincere, it comes to express his contempt and bitterness toward the World State as he grows to know it better. He uses this phrase ironically when he is confronted with dozens of identical and, to him, hideous, dehumanized Deltas.
His passionate quoting of Shakespeare shows how intense and passionate a person he is, and how impossibly far apart he is from the mentality of the World State, so much so that he commits suicide rather than remain a part of it.