What does it mean "given a NFA and its accepted language L build the DFA that accepts L*"?
$L^*$ is the Kleene closure of&nbsp;$L$. This is the language of all strings that can be made by taking a finite number of (not necessarily different) strings from&nbsp;$L$ and concatenating them.  More formally,

$$L^* = \{\varepsilon\}\cup\{w\in\Sigma^*\mid \exists k\geq 1\,\exists w_1, \dots, w_k\in L \text{ such that } w = w_1\cdot\dots\cdot w_k\}\,.$$

Or, if you prefer, for $k\geq 1$, let
$$L^k = \{w_1\cdot \dots \cdot w_k\mid w_1, \dots, w_k\in L\}$$
and then $L^* = \{\varepsilon\}\cup L^1 \cup L^2 \cup L^3 \cup\cdots\,$.

As for how to produce an automaton that accepts that, I suggest you think about it some more yourself, now that you know what it is you're trying to do. A hint would be to use nondeterminism to "guess" where the words $w_2, \dots, w_k$ begin.