Plates move because of gravity, which is the main driving force behind plate tectonics. Another reason why plates move is due to convection.
Convection drives plate tectonics because there are convection cells in the Earth's mantle so that the hot, less dense rock material goes toward the crust of the Earth while the relatively denser and less hot rock material goes toward the core. Convection cells are also responsible for the breaking up of supercontinents throughout Earth's history and were responsible for the East-African Rift. Before the rift, Eastern Europe and Africa were connected.
Convection currents also occur within the atmosphere through wind and within the hydrosphere through ocean currents. It is in the geosphere where they occur as plate tectonics. The aesthenosphere, at depths of approximately 70-250 kilometers, is the area of the inner Earth's magma where heat is generated to build the convection current.
Gravity moves plates because when a plate meets another plate in the oceanic lithosphere, the dense oceanic lithosphere will dive beneath the other plate and sink into the mantle while dragging the rest of the tectonic plate. This creates plate motion and is the main cause of plate motion in the world. This is also the process that creates mid-ocean ridges.