Here's a proof that looks short and snappy. But it has to use a certain amount of machinery. I don't think you'll find a really elementary proof of this result – it certainly isn't obvious.
Fact 1. For any two nxn matrices P and Q, PQ and QP always have the same spectrum (spectrum = set of eigenvalues). There's a short proof of that here.
Fact 2. A positive semidefinite matrix has a positive semidefinite square root . Proof: A can be diagonalised. The square root is obtained by forming the diagonal matrix whose diagonal entries are the square roots of those in the diagonal matrix obtained from A.
If A and B are positive semidefinite then the spectrum of is the same as the spectrum of (by Fact 1). But is positive semidefinite, so its spectrum is real and non-negative.