this is pretty straightforward if you just apply the definitions.
The general way to show two sets are equal is to show they are both subsets of one another, i.e. $A \subset B \wedge B \subset A \Rightarrow A=B$
I'll do one direction and leave the other to you.
let $(x,y) \in (A \times B) \cap (C \times D)$
then by definition $(x,y) \in (A \times B) \wedge (x,y) \in (C \times D)$
then again by definition this implies
$x \in A \wedge x \in C,~~y \in B \wedge y \in D$
thus $x \in A \cap C, ~~y \in B \cap D$ and thus $(x,y) \in (A \cap C) \times (B \cap D)$
thus $ (A \times B) \cap (C \times D) \subset (A \cap C) \times (B \cap D)$
now you do the other direction.