Why do tectonic plates move?
The main driving force of plate tectonics is gravity. If a plate with oceanic lithosphere meets another plate, the dense oceanic lithosphere dives beneath the other plate and sinks into the mantle: this process is called subduction. The sinking oceanic lithosphere drags the rest of the tectonic plate: this is the main cause of plate motion. Oceanic lithosphere is therefore pulled apart in several directions: that process creates the mid-ocean ridges where new, hot and light oceanic crust is created.
However, convection is also driving plate tectonics. When you heat up noodles in a pan of water, you create convection cells: noodles move upward in the middle of the pan where temperature is higher, and downward on the edges of the pan where the temperature is lower. Such convection cells exist inside the Earth's mantle. One difference is that the mantle is not liquid; rather, the solid rocks are so hot that they can slowly flow. Hot, less dense rock material goes toward the crust whereas relatively denser, less hot material goes toward the core.
At certain times and places, hot, upflowing rock material in these convection cells weakens continental crust to create rifts and eventually new ocean basins. The East-African Rift, for instance, is the result of such a convection cell breaking up the African plate. Convection cells were responsible for the breaking up of supercontinents many times in Earth's history.