What causes the Tectonic Plates to move?

The currently accepted theory is that thermal convection is the main driving force. The high temperatures in the earth's interior makes the plastic rocks in the asthenosphere less dense than the rocks above and causes them to rise while the denser rocks sink below. This gives rise to convection currents just like in any other heated fluid. These slow moving currents circulate as shown in the figure below and and exerts drag on the bottom on the lithospheric plate. This frictional force is what causes the plates to move. Furthermore, as the solid and denser plate sinks below into the less dense and plastic asthenosphere it tends to sink further, pulling the rest of the plate downwards. While this is the accepted mechanism at the moment. Further research is yet to provide deeper insight regarding this process.

image credit : Invitation to Oceanography, 5th edition, Paul R. Pinet