The allusion to Macbeth comes when Mustapha Mond and John are debating about this new world that John has entered and the way that "high art" has been sacrificed for stability. Note what Mond says about this when John questions why this society has dispensed with Shakespeare:

But that's the price we have to pay for stability. You've got to choose between happiness and what people used to call high art. We've sacrificed the high art. We have the feelies and the scent organ instead.

Mond therefore argues that the sacrifice of plays by Shakespeare and other forms of "high art" are necessary because of the way in which they allow stability to exist. John responds saying that the feelies are "told by an idiot," which is a direct allusion to Macbeth's speech about life after discovering that his wife has died:

it is a tale
Told by an idiot, full of sound and fury,
Signifying nothing.

Why it is so significant and apt is that it excellently describes the kind of life led by citizens of this new world, as it does signify nothing, because all of the substance has been taken away.