Finding $L^*$ when $L=\{a^nb^n | n \geq 1\}$
$L^2 = LL = \{uv | u,v\in L\}$. In words, $L^2$ is a set of all strings that formed by concatenation of all strings from $L$. For instance, if $u=a^kb^k \in L$ and $v=a^tb^t \in L$ then $uv = a^kb^ka^tb^t$. That's why $L^2 =\{ a^{n_1}b^{n_1}a^{n_2}b^{n_2}\}$ for all $n_1, n_2 \geq 1$.
Similarly $L^3, L^4,\dots$

Example with a finite set: $L=\{ab, cd\}$. Then $L^2 = \{abab, abcd, cdab, cdcd\}$.