Dystopian novels typically will present a character or an element of the story that represents "the past"—the world and the "values" of the world as they existed prior to the cataclysm or transformation that resulted in the dystopia.
In Orwell's 1984, the elderly man Winston talks to in the pub, and then later Mr. Charrington's shop, functions this way, though the latter turns out to be an illusion when Charrington's actual identity is revealed. In Ayn Rand's Anthem, it's the forbidden forest to which the couple escape in the end. In Brave New World, it's the "Savage Reservation," and specifically John "the Savage," that represents the pre-dystopian world as foils to the Society and its sterile modernity.
Why, though, should John's knowledge of Shakespeare in particular be the mechanism by which the old world is given to us? Probably because most people, in Huxley's time as in our own, would regard Shakespeare as the most iconic author of our "modern" times—meaning from the Renaissance forward. As Orwell wrote, it's difficult for people who have read Shakespeare not to quote him often, because Shakespeare had so much to say on so many different subjects.
In his constant quotations of Shakespeare, John serves as an emblem of the phase of human history in which Shakespeare is so highly regarded and seen as a kind of universal commentator on every aspect of life. If John sees the world through Shakespeare, it is Huxley's way of enabling us to see the ethos and the standards of the pre-dystopian society—the society which is that of Huxley's and our own time as well, but one which, in Brave New World, Huxley predicts the demise of.
Though Huxley may not have intended it quite this way, the transference of Shakespeare's standards and language onto John is a double-edged sword. The ultra-poetic language is an antidote to the harsh, empty reality of the future. But it also preserves the old, judgmental, and destructive standards of the past. John's yelling at Lenina that she is a "strumpet" when she comes on to him is one of the more unfortunate and negative legacies of the past: gender stereotypes and the abuse towards women.
The very name of the book (and the way John enunciates it in the narrative) encapsulates a dual meaning of positive and negative, good and bad. The obvious intent of the title is an ironic one—this is not a "brave" (good) new world at all but the opposite. Yet one has to wonder if Huxley (although he later, in Brave New World Revisited, refers to his famous vision as a "nightmare") is possibly using the quote from Shakespeare equivocally, as a kind of ambiguous test of the reader's reaction to this projection into the future of the new, mechanized, and spiritually barren aspects of the twentieth-century world, which nevertheless might be at least partially an improvement over the worst things in the past and present.