There is a simple proof of Frobenius's theorem in Lam's book on noncommutative rings, pp. 208--209. He attributes the argument to Palais.
One should consider this theorem to be two theorems: (1) C is the only C-central division algebra and (2) R and H are the only R-central division algebras. The reason there are so few choices is that C is alg. closed and R is nearly so. Division algebras with center equal to a particular field can be created using cyclic Galois extensions* and since Q has such extensions of arb. high degree there are Q-central division alg. of arb. high dimension.
*There are further technical conditions to be satisfied on the cyclic extension in order for the construction of a division algebra to work, e.g., a finite field has a cyclic extension of each degree but there are no central div. alg. of dim. > 1 over a finite field. The relevant technical conditions are satisfied when the base field is the rational numbers.