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) $\mathbb C$ is the only $\mathbb C$-central division algebra and (2) $\mathbb R$ and $\mathbb H$ are the only $\mathbb R$-central division algebras. The reason there are so few choices is that $\mathbb C$ is alg. closed and $\mathbb R$ is nearly so. Division algebras with center equal to a particular field can be created using cyclic Galois extensions* and since $\mathbb Q$ has such extensions of arb. high degree there are $\mathbb Q$-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.