Kleene positive closure - help in proofing this claim
Let's use $A$ for $L_1$ and $B$ for $L_2$.
If you unpack both sides, you get
$$
(AB)^+ = AB + ABAB + ABABAB + \cdots \\
A^+B^+ = AB + AAB + ABB + AABB + \cdots
$$
Perhaps this can give you some ideas.