Let $X$ be a set. What is $X\times \emptyset$ supposed to mean? Is it just the empty set?
And more can be said: a cartesian product is empty if and only if one of the two factors is empty.
$X \times \emptyset = \emptyset$. In fact, if not we have $x \in X $ and $y \in \emptyset$ such that $(x,y) \in X \times \emptyset$. But $y \in \emptyset$ is impossible.
Recall the definition of $A\times B$: $z\in A\times B$ if and only if $z=\langle a,b\rangle$ where $a\in A$ and $b\in B$. That is $A\times B$ is the set of all ordered pairs whose first coordinate is in $A$ and second in $B$.
If $B$ is empty then there are no ordered pairs $\langle a,b\rangle$ such that $b\in\varnothing$, therefore $A\times\varnothing=\varnothing$. Similarly $\varnothing\times B=\varnothing$.