You can split a matrix product like $AB$ according to the columns of the right factor $B$: each column of $AB$ is equal to $A$ applied to the corresponding column of $B$ (this is how matrix multiplication works; you can also split the product according to the rows of $A$, but that is not helpful here).
So saying $AB=0$ means precisely that for each column $c$ of $B$ you have $Ac=0$, or $c\in\operatorname{null}A$. So you have to prove the column space of $B$ is contained in $\operatorname{null}A$ if and only if each individual column of $B$ lies in this same space $\operatorname{null}A$. You should be able to prove that yourself.