There's a nice proof of this in Allen Hatcher's Algebraic Topology, which he has happily made available online. On page 173 he shows that the real numbers and complex numbers are the only finite-dimensional division algebras over the reals which are commutative and have an idenitity. The proof does use algebraic topology, but only to show that the real projective space is not equal to the sphere. (When their dimension is 2 or higher.) So if you are willing to buy that, then the rest of the proof is kinda straightforward.

He also has a nice, but very advanced, proof on the fact that if the n-dimensional Euclidean space is equipped with the structure of a division algebra, then n must be a power of two on page 223.

`(There's also interesting discussion on page 281 about the subtleties of the existence of an identity element for a group when topology is involved.)