Let $A$ where it's non-zero singular values are $\sigma_1,\ldots,\sigma_r$. $A$ can be written as $A=U\Sigma V$ where both $U,V$ are unitary matrices.
Let's look at $$AA^T = U\Sigma V^TV\Sigma U^T= U\Sigma^2 U^T$$
Now, from this point one should infer that the eigenvalues of $AA^T$ are $\sigma_1^2,\ldots,\sigma_r^2$ but I'm not sure how exactly. Is that has something to do with the fact that $U$ is unitary?