A proper subspace of a normed vector space has empty interior clarification

What this means is that $L^\infty$, as a subset of $L^1$, has empty interior (since $L^\infty$ is a proper subspace of $L^1$), as you have stated.

But $L^\infty$ is not nowhere dense in $L^1$. In fact, $L^\infty$ is dense in $L^1$. This is because any $L^1$ function can be approximated by simple functions, which are in $L^\infty$. Thus the closure of $L^\infty$ is $L^1$, which certianly has nonempty interior in $L^1$.