The proofs certainly exist, although I haven't looked at them in much detail. Higher-dimensional geometry gets pretty complicated pretty quickly, as you can imagine.

It wasn't until the mid-late 19th century that people seriously did geometry in dimensions higher than 3-4; this Wikipedia page gives decent historical overview. Coxeter's Regular Polytopes is THE book on the matter, I'm certain it's in there (incidentally, Coxeter is the guy that beat me to this result in the 1930s. But from the literature review I've done, very few people noticed this particular result, so it's more-or-less entirely unknown among the mathematical community).

But you can give a rough sketch of why those 3 exist without too much effort. I'll just describe them, and only say a bit more about about complicating factors.

The entire framework of coordinate geometry is basically premised on the existence of cubes. If you look at the vertices of the 3-cube you can see how it should work in higher dimensions:

           (0,0,0)

(1,0,0)    (0,1,0)    (0,0,1)

(1,1,0)    (1,0,1)    (0,1,1)

           (1,1,1)

All vertices have coordinates either 0 or 1, and there's a vertex for each possible combination. The cube is the "convex hull" of these vertices (like starting with the vertices and shrink-wrapping around them). They're gonna have 2^n vertices and 2n facets in dimension n ("facets" = faces of the highest dimension. Eg, six 2-dimensional faces in 3D). You can know everything about its faces of various dimensions if you're good with combinatorics, because the n-cube has a strong relationship with the 2^n subsets of an n-element set. For any computer science nerds out there, Karnaugh maps are just exploiting this relationship between subsets and cubes.

For the cross-polytope (aka orthoplex), it's similarly based on coordinate axis. Its vertices in 3D are even easier:

(1, 0, 0)    (0, 1, 0)    (0, 0, 1)

(-1, 0, 0)   (0, -1, 0)   (0, 0, -1)

In general, these guys are the convex hull of plus-or-minus the standard basis vectors. They have 2n vertices in dimension n, and 2^n facets (triangles in 3D). That's because cubes and cross-polytopes are dual to one another. That wikipedia page is specifically about 3D, but the concept holds in any dimension. So cubes and cross-polytopes exist in all dimensions as a tight pair: if you believe in one, you have to believe in the other.

Then there's the simplex (triangular pyramid in 3D). It's a little less well-known, but unequivocally simpler: they have n+1 vertices and n+1 facets in dimension n.

It's easier to jump up an extra dimension to find them. There's a 2D one hiding among the vertices of the 3-cube:

(1, 0, 0) (0, 1, 0) (0, 0, 1)

So, it's relatively easy to describe the shapes and give realizations of them. The hard part really comes talking about "regularity". In 2D, it's easy: Equal edges, equal angles. We give an iterative process for 3D: Regular faces, all faces congruent, and the same number meeting at each vertex. But how to generalize into 4D? 5D? Do we need to keep adding onto the definition each time?

So that way of thinking was judged insufficient. The whole point is that we want shapes that look “the same” from multiple viewpoints, to the highest extent possible. Thus, the current state-of-the art is to define regularity by saying that "A polytope is regular if and only if its symmetry group acts transitively on its set of flags", where a flag is a complete, nested sequence of faces (for example: A vertex contained in an edge contained in a face. If you pick two of those for the 3-cube, you can find a way to spin/reflect the cube and swap those two flags). So now you've really gotta be at least somewhat comfortable with group theory, and a whole host of other math kinda slides in at this point. It's not too bad I don't think, but it takes work.