Proving a sum of matrices is invertible
To show that $M^{-1}$ is the inverse of $M$, it suffices to show that $M^{-1}M = I$.
To that end, note that $$ [I + B(I - AB)^{-1}A](I - BA) = \\ I - BA + B(I - AB)^{-1}A - B(I - AB)^{-1}ABA =\\ I - BA + B[(I - AB)^{-1} - (I - AB)^{-1}AB]A =\\ I - BA + B[(I - AB)^{-1}(I - AB)]A =\\ I - BA + BA = I. $$ So indeed, $I + B(I - AB)^{-1}A$ is the inverse of $(I - BA)$.