I am not sure if this helps when teaching a basic PDE class, but this is certainly a useful understanding:
Elliptic problems can be interpreted via diffusion processes. The solution at a point $x$ can be written as expectation of the boundary condition at the (random) exit point for the diffusion emitted from $x$ and associated to the elliptic operator.
If the boundary is smooth, then as $x\to x_0\in\partial \Omega $ the exit distribution converges to the Dirac measure at $x_0$, hence regularity of the solution.
If the boundary is bad, then the diffusion initiated at a boundary point $x_0$ can, with positive probability, hit the boundary next time at a completely different place, and the exit distribution can be very far from the Dirac measure, hence there is a problem.