Second Hochschild cohomology and extensions

Question

I started learning the theorem that says there is a one-to-one correspondence between $\mathrm{Ext}(A, M)$ and $H^2(A, M)$. However, the proof is not clear. I managed to show that there is a well-defined map $U$ from $\mathrm{Ext}(A, M)$ to $H^2(A, M)$ and a well-defined map $V$ from $H^2(A,M)$ to $\mathrm{Ext}(A, M)$. But, I am really stuck to show that these two maps are inverse to each other? So, any help to do that, please? Thank you.