First thing you should do is write the definitions:
- $\ker\psi=\{v\in V\mid \psi (v)=0\}$.
- $\ker(\psi\circ\psi)=\{v\in V\mid \psi(\psi(v))=0\}$.
- $A\subseteq B$ if and only if for all $a\in A$, $a\in B$.
Next you should note that we always have $0\in\ker\psi$ when $\psi$ is linear. Now this is amounts to a standard element chasing proof:
Let $x\in\ker\psi$, we want to show that $x\in\ker(\psi\circ\psi)$, namely $\psi(\psi(x))=0$. However since $x\in\ker\psi$, and $\psi$ is linear we have that $\psi(\psi(x))=\psi(0)=0$ Therefore $x\in\ker(\psi\circ\psi)$ as wanted.