I am following Applied statistical decision theory [by] Raiffa, Howard. Which can be consulted online here.
A theorem at the page linked states that if two matrices $A,B$ are non-singular and of dimension $r \times r$, then:
$$(A+B)^{-1} = B^{-1}(B^{-1}+A^{-1})A^{-1} = A^{-1}(A^{-1}+B^{-1})B^{-1} $$
Why is this true?
(a link to a proof will suffice).
$$(A+B)^{-1}=[B(B^{-1}A+I)]^{-1}=[B(B^{-1}+A^{-1})A]^{-1}$$
Now use that $(XY)^{-1}=X^{-1}Y^{-1}$ to get
$$[B(B^{-1}+A^{-1})A]^{-1}=A^{-1}(B^{-1}+A^{-1})^{-1}B^{-1}$$
Notice that there is a missing $-1$ in your formula.
To get the other equation start with
$$(A+B)^{-1}=[A(I+A^{-1}B)]^{-1}=[A(B^{-1}+A^{-1})B]^{-1}$$ and then
$$[A(B^{-1}+A^{-1})B]^{-1}=B^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}$$