This is proved in the book *Representations of compact Lie groups* by Bröcker and tom Dieck and reviewed [here][1].  It is Proposition 7.5 in Chapter V.  The proof is for compact, connected Lie groups, but any connected Lie group has the homotopy type of its maximal compact subgroup.  (Everything here is for finite-dimensional Lie groups, of course.)

**Edit**: Maybe I should add, given that two of the other answers mention similar proofs, that the one in this book does not use Morse theory.  It uses only basic covering space techniques once it is shown that $\pi_2(G)$ is isomorphic to $\pi_2(G_r)$, where $G_r$ are the regular elements, itself not a difficult lemma.

Also if you believe that simply-connected compact Lie groups have the homotopy type of a product of odd spheres, then this follows.  It is fairly easy to see that this is the case rationally, but do not remember whether the general statement is hard to prove.

  [1]: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183553688