Can someone explain the language L = {w: w = uu, u \in La(1*01*)}
No, your simplification is wrong.

Consider, as a simpler case, $A=\{a,b\}$ and $L'=\{ w \mid w=uu , u\in A\}$. Then, $L'=\{aa,bb\}$.

Instead, if we apply your simplification and write $L''=AA$ then we have $L''=\{aa,ab,ba,bb\}$ which is a larger language, since we forgot that both $u$'s in $w=uu$ must be the same word.

Keeping that into account, your original language $L$ seems to be $\{1^n01^m1^n01^m \mid n,m\geq 0 \}$