I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem.
It says, if $u(x)$ is harmonic on $\Omega$ and boundary $\partial\Omega $ of $\Omega$ is smooth then $$ \int_{\partial \Omega }\frac{\partial u(x)}{\partial n}ds=0, $$ where $n$ is a normal vector I guess. This clearly says there is no flux.
What is this theorem really about and how can I find the proof?
The heat equation is $\alpha \Delta u = \partial_t u$. So if $u$ is harmonic, $\nabla^2 u = \Delta u = 0$, we are saying the system is in equilibrium, $\partial_t u = 0$, and thus there is no net heat flow in or out of the region $\Omega$.
Proving this result uses an equivalent form of the directional derivative, the divergence theorem, and the fact that $u$ is harmonic:
$$ \int _{\partial \Omega }\frac{\partial u}{\partial n}ds = \int _{\partial \Omega } \nabla u \cdot n \ ds = \int_\Omega \operatorname{div}(\nabla u) \ dV = \int_\Omega \Delta u \ dV = \int_\Omega 0 \ dV = 0 $$