Take a <context-free grammar> for $L$ with start symbol $S$, rename nonterminals if necessary, and add a fresh start symbol $S_0$ with
$$
S_0\to\epsilon\mid SS_0.
$$
Every derivation concatenates finitely many words of $L$, and every finite concatenation can be derived. \b[The resulting grammar generates $L^*$], proving <closure of context-free languages under Kleene star>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 4G / c / Solution

If a <context-free grammar> for $L$ has start symbol $S$, adjoining a new start symbol $S_0$ and productions $S_0\to\epsilon\mid SS_0$ gives a context-free grammar for the <Kleene star> $L^*$.

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 Incoming links: Closure of context-free languages under Kleene star