Tor functor,deleted last term,exactness

Question

Let have a setting like this on page 28. I have some questions on the deletion of the last term $M\otimes N$. We loose by this isomorphism Tor$_0^R(M,N)\cong M\otimes_R N$. The second is why the deleted tensored sequence is exact at the last arrow $\to P_0\otimes N\to 0$? Finally, why did we at all erase the last term $M\otimes_R N$ in front of $0$?