In the process of calculating a fractal such as the Mandelbrot set, a simple equation is applied repeatedly to each pixel in the fractal window. The equation takes the coordinates for that pixel, does some math, and produces another pair of numbers to serve as the input to the next iteration. This pair of numbers can be thought of as being a set of coordinates. If the point is a member of the Mandelbrot set, these points will stay forever quite near to the origin.
One way that a sequence of points, or calculation path, can be sure that it never strays too far from home is to run around in circles. That is, to loop around a bit and then start following in its own footsteps. Once a point in a calculation sequence returns to a point it has previously visited, it is trapped. From then on, it will always follow exactly the same sequence of steps, running around the same sequence of points forever more.
That's one way that a point can avoid running away to infinity, and it turns out (I believe) that it is the only way. Points either get trapped in a cycle or run around for a while and then head for the hills.