Let $\Omega$ be a domain which may or may not be unbounded (eg. $\Omega = B_1(0)\times (0,\infty)$). Why is $$ \overline{L^\infty(\Omega)} \subset L^\infty(\Omega)$$ where the closure is in norm of $L^1(\Omega)$?
I would have thought that the inclusion is the other way around. Source is this paper, see page 11, the third displayed equation.
As written, this is either false or trivial, or both. The closure of any subset of $X$ in $X$ is trivially a subset of $X$; just because of what the word closure means. Closure is taken with respect to a particular ambient space, and cannot produce elements outside of that space.
If you meant completion with respect to $L^1$ norm, then the inclusion is false. The completion of $L^\infty$ with respect to $L^1$ norm can be identified with $L^1$.
So, you misunderstood something. The only conclusion from here is:
Give the source of statements that you would like explained to you.