How to prove that $L^* = L^*L^*$
Here is a partial answer, to put you on the right track. Try to complete it to a full answer.
Let's try to show that $L^*\subseteq L^* L^*$.
Let $x\in L^*$, so we know that we can write $x=y_1\cdots y_k$ where $y_i\in L$ for all $i$.
Now, let's assume for a moment that $k\ge 2$, then we can look at two words: $x_1=y_1$ and $x_2=y_2\cdots y_k$. We then have that $x=x_1x_2$ and $x_1\in L^*$ and $x_2\in L^*$, so $x\in L^*L^*$.
If $k&lt;2$, then we can take $x_2=\epsilon$, and possible $x_1=\epsilon$, and the proof still works. So we have $L^*\subseteq L^* L^*$.
Can you now follow a similar scheme for the other containment?