From Introduction to Automata Theory, Languages, and Computation (2nd Edition):
"For a final example, the closure of the empty set = { epsilon } where epsilon is the empty string. Empty set is one of only two languages whose closure is not infinite." What is the other language?
The Kleene closure is an idempotent operator: given a language $L$, it holds $(L^*)^*=L^*$. So another such language is...
For the converse, suppose $\sigma\in L$ and $\sigma\ne\epsilon$. Then $\{\sigma,\sigma\sigma,\sigma\sigma\sigma,\cdots\}\subseteq L^*$ and all those strings are distinct, because they have strictly increasing length.