By using the contrapositive, we can write the LHS as:
$$(B \rightarrow A) \land (A \rightarrow C)$$
This says that if $x \in B$, then $x \in A$ and if $x \in A$, then $x \in C$. Both of them hold true together.
We want to prove that if $x \in B$, then $x \in C$.
Let us assume the LHS is true.
Now, if $x \in B$, then by LHS we know $x \in A$. Now since $x \in A$, $x \,\text{also} \in C$, as that's what we have assumed to be true.
This shows that if $x \in B$, then $x \in C$.
You can look at it as a sort of transitive relation.
