Given a set V define
$V_0 = \left \{ \epsilon \right \}$ (the language consisting only of the empty string),
$V_1 = V$
and define recursively the set
$V_{i+1} = \left \{ wv : w ∈ V_i \ \text{and} \ v \in V \right \} \text{for each i>0.} $
If V is a formal language, then Vi, the i-th power of the set V, is a shorthand for the concatenation of set V with itself i times.
The definition of Kleene star on V is
$V^* = \bigcup\limits_{i\geq 0}^{} V_i = {\epsilon} \ \cup \ V_1 \ \cup V_2 \ \cup \ V_3 \ \cup .....$
Now coming to language provided in question,
$L = \{ 0^n 1, \ n \geq 0 \}$
$L^* = \{ 0^n 1, \ n \geq 0 \}^* $
$L_0 = \{ \epsilon \}$
$L_1 = L = \{ 0^n 1, \ n \geq 0 \}$
$L_2 = \{ wv : w \in V \ \text{and} \ v \in V \}$ = $\{11, 101, 0101, .... \}$
and so on for $L_3 , L_4 ,....$
$L^* = \bigcup\limits_{i\geq 0}^{} L_i = {\epsilon} \ \cup \ L_1 \ \cup L_2 \ \cup \ L_3 \ \cup .....$
this can also be written as $(0^*1)^*$ and it generates a regular language.
