In chapter 6 of Eisenbud's book describes the next propertie of Tor functor:
As in the case of $\operatorname{Tor}_0$, $\operatorname{Tor}_i^R(M,N)$ is a covariant functor of two $R$-modules $M$ and $N$ that is $R$-bilinear in the sense that it is an $R$-module, and the map "multiplication by $r \in R$" applied to either $M$ or $N$ induces "multiplication by $r$" on $\operatorname{Tor}_i^R(M,N)$.
But he doesn't explains how this multiplication works. Someone can explain me?
For $r \in R$, you have a linear map $\mu_r : M \to M$ given by "multiplication by $r$": $\mu_r(m) = m \cdot r$. Since $\operatorname{Tor}_i^R$ is covariant, this induces a map $$(\mu_R)_* : \operatorname{Tor}_i^R(M,N) \to \operatorname{Tor}_i^R(M,N).$$
Similarly for $N$, you have the linear $N \to N$ given by $n \mapsto r \cdot n$ which induces the same sort of map. Now what Eisenbud is saying is that $\operatorname{Tor}_i^R(M,N)$ is an $R$-module, so given $x \in \operatorname{Tor}_i^R(M,N)$ and $r \in R$, there is a given element $r \cdot x \in \operatorname{Tor}_i^R(M,N)$. Then it so happens that $(\mu_R)_*(x) = r \cdot x$.