I've read that, given a bipartite pure state $|\Phi\rangle$, its entanglement (equivalently here, von Neumann) entropy $E(\Phi)$ gives the asymptotic number of singlets required to create $n$ copies of $|\Phi\rangle$. More specifically, that for all $\epsilon>0$ there is a large enough $n$ such that, given $m$ copies of a singlet state $|\Phi^-\rangle$, Alice and Bob can produce $n$ copies of $|\Phi\rangle$ with $m/n\le (1+\epsilon)E(\Phi)$. I think this roughly means that $E(\Phi)$ is the number of singlets required to generate each copy of $|\Phi\rangle$.
This is mentioned in this paper by Wooters: http://www.rintonpress.com/journals/qic-1-1/eof2.pdf, around equation 3.
To back up this statement, they cite this paper by Bennett and collaborators. I know that the above statement is shown in this paper, but I was wondering if there was a way to briefly sketch the reasoning, or more generally a way to understand the intuition behind it. More recent takes on this result would also be welcome, if there is any.