Definitions 7.7.14: Let $f : A \to B$ be a ring homomorphism. Let $M$ be a $B$-module. The $A$-module structure on $M$ induced by $f$ is given by the structure map $a \cdot m = f(a)m$ for $a \in A$ and $m \in M$. Let $N$ be an $A$-module. We say that a $B$ module structure on $N$ is compatible with the original $A$-module structure if the $A$-module structure induced by $f$ from the $B$-module structure is the same as the one we started with (i.e., $f(a) \cdot m = am$ for all $a \in A$ and $m \in M$).
I think I understand the first part: it's essentially showing us how to define $M$ to be a $A$-module, besides just being a $B$-module; multiplication like $am$ by itself doesn't make much sense, so we'll just define multiplication by $a \in A$ with $m \in M$ as $f(a)m$. However, I am having trouble parsing the bolded part, especially on how it relates to the unbolded part.
Say that $N$ is both an $A$ module and a $B$ module. Then the fact that we have $f:A\rightarrow B$ means that we have a second $A$-module structure on $N$ as described in the unbolded part of your definition. This second $A$-module structure depends on what the $B$-module structure of $N$ is. If these two $A$-module structures happen to coincide (ie, $a\cdot n=f(a)\cdot n$), then this means that the $B$-module structure does not interfere with what $A$ was already doing. In this case, we say that the $B$-module structure is compatible with the original $A$-module structure.