The Tempest first pops up in the text among Mrs. Who's plethora of quotations:
"We are such stuff as dreams are made on." She smiled broadly. "Prospero in The Tempest. I do like that play." (5.55)
But what sets this allusion apart from everything else Mrs. Who says is that it keeps coming back. When Mrs. Who leaves the children on Camazotz, she gives Calvin another passage from The Tempest as "a hint":". . . For that he was a spirit too delicate
To act their earthy and abhorr'd commands,
Refusing their grand hests, they did confine him
By help of their most potent ministers,
And in their most unmitigable rage,
Into a cloven pine; within which rift
Imprisoned, he didst painfully remain. . . ." (6.87)
Besides being good stuff, what significance does The Tempest have for our heroes? You should really go read the whole thing (or at least check out the Shmoop guide), but here's a quick summary of the plot: a group of sailors are shipwrecked on an island inhabited by the powerful magician and banished duke Prospero, his daughter Miranda, and his servant, the spirit Ariel (he's the one in the lines above; he was trapped in a tree by a witch, and Prospero released him from that imprisonment but also enslaved him at the same time). Prospero uses his powers to manipulate the sailors according to his whims, and hilarity ensues.
What does all this mean for A Wrinkle in Time? Well, for one thing, the island in The Tempest is a place full of illusions, just as the world of the novel is. It's run by an all-powerful sorcerer, Prospero, whose intentions are somewhat better than IT's, but who still has a tendency to meddle with the free will of others. And of course it has Ariel, imprisoned in the cloven pine, who, as even Meg recognizes, bears a striking resemblance to her father, locked inside a glass column on Camazotz for not doing what IT wants him to do.
OK, so there are a bunch of parallels, but so what? Why bother? Perhaps the Shakespearean underpinning is there to show that Mrs. Who's language of allusions is a feature, not a bug. Or perhaps it's there to show how the list of members of the Anti-Darkness Justice League that Calvin, Charles Wallace, and Meg come up with are still contributing to the good fight even after their death. Or perhaps it's there to say, "Hey, this may be a kids' book, but we can quote Shakespeare with the best of them, so take us seriously, OK?" Whatever the purpose, the book's conversation with Shakespeare provides an interesting undercurrent to its own unique story.