If you may appeal to the result that (MN)^{-1} = N^{-1}M^{-1} for any invertible matrices M and N of equal size, then the statement you are trying to prove is an immediate consequence:

A(A+B)^{-1}B

= (B^{-1}(A+B)A^{-1})^{-1}

= (B^{-1}AA^{-1} + B^{-1}BA^{-1})^{-1}

= (B^{-1}I + IA^{-1})^{-1}

= (B^{-1} + A^{-1})^{-1}.

If you need to first prove that (MN)^{-1} = N^{-1}M^{-1}, then just multiply MN by N^{-1}M^{-1} and see what you get!