If $L$ is a $CFL$, then why isn't $L^*$ also $CFL$
In fact, if $L$ is context-free then so is $L^*$. The easiest way to see that is using context-free grammar. Every context-free language $L$ is generated by some context-free grammar $G$ with starting symbol $S$. Now add a new starting symbol $S'$ and the productions $S' \to S'S|\epsilon$. The new grammar $G'$ is also context-free and generates $L^*$.