I can't comment yet so let me post as answer. Forget about the coefficients and just take $L=-\Delta$, the Dirichlet Laplacian. Then you are asking why $-\Delta u \in H^{-1}(\Omega)$ when $u \in H^1_0(\Omega)$. It is because the weak Laplacian is defined $$\langle -\Delta u, v \rangle_{H^{-1}, H^1_0} := \int_\Omega \nabla u \nabla v.$$ It is easy to verify that this operator $-\Delta u$ is a bounded linear functional belonging to the dual space of $H^1_0$.
So it seems incorrect in your OP to say " I can't figure out this simple fact." because it is a definition.