The best way to understand it is to think of a diffusion problem. We have some kind of source, perhaps more than one, leaking out molecules. Each molecule bounces around randomly until it reaches the edge, then escapes. This steady-state concentration gradient is described by Poisson's equation. The region the molecules diffuse around in is represented by a grid of points.
Diffusion is simulated by a random walk.
At every step, a molecule steps to one of the four adjacent points. Eventually it steps off the edge, and the walk ends. The program keeps track of the total number of times each point was hit in the random walks. One the average this is proportional to the steady-state concentration, which solves the equation. The other way to solve Poisson's equation, and the most commonly used, is to write down Poisson's equation for each point, then solve all of them together.