Choose $\vect{x}\in\csp{AB}$. Then by Theorem CSCS there is a vector $\vect{w}$ that is a solution to $\linearsystem{AB}{\vect{x}}$. Define the vector $\vect{y}$ by $\vect{y}=B\vect{w}$. We are set,
\begin{align*}
A\vect{y}
&=A\left(B\vect{w}\right)&&\text{Definition of $\vect{y}$}\\
&=\left(AB\right)\vect{w}&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\
&=\vect{x}&&\text{$\vect{w}$ solution to $\linearsystem{AB}{\vect{x}}$}\\
\end{align*}
This says that $\linearsystem{A}{\vect{x}}$ is a consistent system, and by Theorem CSCS, we see that $\vect{x}\in\csp{A}$ and therefore $\csp{AB}\subseteq\csp{A}$.
For an example where $\csp{A}\not\subseteq\csp{AB}$ choose $A$ to be any nonzero matrix and choose $B$ to be a zero matrix. Then $\csp{A}\neq\set{\zerovector}$ and $\csp{AB}=\csp{\zeromatrix}=\set{\zerovector}$.