Seismology: What causes an earthquake?
A lot of stress is accumulated along the fault by the time it breaks. For example, on the San Andreas, the Pacific plate is moving north with respect to the North American plate, but the fault resists this motion. As a result, a ton of elastic energy builds up around the fault as the stress increases. 

Eventually, the stress becomes so much that the fault slips. The fault slipping releases the stored elastic energy, some of which is radiated as seismic waves. (Most of the energy becomes heat.) 

Seismic waves "carry" stress and motion. When a wave first arrives, the rock at that location experiences a change in stress. This causes the rock at that spot to move slightly. But then, the rock at that spot has moved with respect to nearby rock. This differential motion of the rock creates stress, which then affects nearby rock similarly. So, the wave's stresses propagates motion in the rock, and motion in the rock propagates stress. 

A few types of waves exist (P waves, S waves, and surface waves, in descending order of wave propagation speed). But the general idea behind them all is the same.

The ground therefore shakes when the waves arrive.