This question is best asked with a picture:
In words, we can flatten a cube into 2-d space and get a set of flattened squares like in the top right of the picture where five of the edges have stayed intact (the squares connected by them are still connected). It should similarly be possible to flatten a Teserract into 3-d space and get a set of cubes that are connected to each other. What will this picture of the flattened Teserract look like?
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There are, of course, many ways to unfold the hypercube (or for that matter, a three-dimensional cube). Here is one way:
Think of a ring of four square faces in a three-dimensional cube. If you break the ring you get a chain of four squares, onto which you can put the remaining two faces anywhere as long as they are on opposite sides of the chain. You can form a similar chain with four "outer" cubes in the standard projection of the hypercube, and place one of the remaining cubes on each lateral surface of the chain. https://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus) shows one such configuration, rendered into art by Salvador Dali, which is interpreted as a higher-dimensional version of the Cross from Christianity.
There are, in fact, 261 different ways to unfold the tesseract into 3D space, as proved in this reprint of a paper from Journal of Recreational Mathematics, Vol. 17(1), 1984-85. As far as I know, I'm the first person to actually use the ideas in that paper to generate all those 3D models, as described in this mathoverflow post. All the images are shown below; Dali's image that Oscar refers to is in the middle of the third row from the bottom.