What the heck is a tesseract? We'll let Charles Wallace explain:
"Well, the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points." (5.36)
Technically the tesseract is four-dimensional, not five, but that's a quibble. (Still confused about what a tesseract is? See the "Best of the Web" section for more attempts to explain the tesseract in a way that makes sense, with pictures and animations even.) The point of bringing the tesseract into the book isn't to offer a course in Hypercubes for Dummies, but rather to suggest that the novel's fantasy is grounded in science.
But why does it matter whether there's a scientific basis for the crazy stuff that goes down in the book or not? Throwing in a few words that you'll find long complex explanations for on Wikipedia doesn't make you any more likely to believe that somewhere in space there's a planet full of winged centaurs with an impossibly beautiful list of Top 40 hits, so why bother? Perhaps L'Engle is trying to balance fantasy with fact, so that even her weirdest creations seem more believable. Or perhaps she's pointing out that even fact can be fantastical: if the real world has such bizarre concepts in it, perhaps we don't need winged centaurs to see reality as wondrous and awe-inspiring.
So if you don't understand the tesseract, never fear: it's all part of the mysterious grandeur of the universe.