Number of words in $L$ of length $n$?
Means that $|L^n|$ is the number of words in $L$ of length $n\in \mathbb{N}$ letters from $\Sigma$.


Actually, $L^n$ consists of the words that are composed of $n$ copies of words from $L$. The precise lengths of these words differ.

This exercise starts however with a very specific set $L$, it consists of all words of length 1 or 2. That means that $L^2$ consists of all words of length 2,3 or 4. (No need to verify in how many ways that can be done.) And this are 4+8+16 = 28 strings, exactly like you claimed.

This can be generalized for $L^n$.