The statement is true. The proof just uses the relevant definitions.
Suppose $x \in A \cap \operatorname{cl} (B)$, i.e., $x \in A$ and $x \in \operatorname{cl}(B)$. The latter condition means that if $U$ is an open set containing $x$, then $U$ intersects $B$ (nontrivially).
Now, pick any open set $V$ containing $x$. Then $V \cap A$ is an open set containing $x$. Then by the preceding paragraph, $(V \cap A) \cap B \neq \emptyset$, which is equivalent to $V \cap (A \cap B) \neq \emptyset$. This means that $x \in \operatorname{cl}(A \cap B)$.