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I was given a following statement and was told it is true, but I can't figure out why. Here it is.

Let's say we have a following matrix product, $ C=AB $ Null space of $C$ is a subspace of null space of $B$.

Can someone explain why this is the case. I can't figure it out.

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As Andreas Caranti points out in his answer, the statement is not true, and in fact it’s the other way around: $\operatorname{Null}(B)\subseteq\operatorname{Null}(C)$. It’s not hard to see why the null space can only grow, not shrink, when you multiply by another matrix. Multiplying the zero vector by anything gives the zero vector, so if $B$ sends a vector to zero, it stays there when you multiply by $A$. So, multiplying by $A$ can add more vectors to the total null space of the product, but it can’t remove any.

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    I'm not sure I understand. B contains vectors that make A vectors equal to 0. How is it related to matrix C, what is the connection? – 2017-02-17
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    The null space is the set of vectors that are “collapsed” to zero by a matrix. Once they’ve collapsed, they stay collapsed: any matrix times zero is still zero. Multiplying the results by another matrix can only add to the set of vectors that have collapsed. It can’t get any smaller. Thus it can only be the case that $\operatorname{Null}(B)\subseteq\operatorname{Null}(AB)$ for *any* matrices $A$ and $B$. – 2017-02-17
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It's the other way around, as you can see by taking for instance $B$ to be the $n \times n$ identity matrix (so the null space is $\{ 0 \}$), and $A$ to be the zero $n \times n$ matrix (so that $C$ is also zero, and thus its null space is the whole space).

If $B v = 0$, then $C v = (A B) v = A (B v) = A 0 = 0$.