Suppose we have the following free resolution of a module $N$ $$0\dots\to F_2\to F_1\to F_0\to N\to 0 $$
By definition, this free resolution is exact. Now we tensor this by a module $M$. We get $$\dots\to M\otimes F_2\to M\otimes F_1\to M\otimes F_0\to M\otimes N\to 0$$
Why is $Tor_0^R(N)=M\otimes N$?
The kernel of $\partial: M\otimes F_0\to M\otimes N$ cannot be the whole of $M\otimes N$ right?