So the reason that warm rock is warm is because all of the particles that make up the rock have lots of KINETIC energy. The kinetic energy in the particles causes them to move around a lot and spread out.
Now because the particles are more spread out, there is a lower mass per volume ratio, which is just a fancy way of saying that the warm rock is LESS DENSE.
Now at the same time, the cool rock at the top, which is further away from the molten core, has used up all the kinetics energy, so the particles are slowing down. This means that the particles pack more closely together and so the cooler rock is MORE DENSE.
Okay, now imagine when you drop a rock in water; it sinks to the bottom. That's because it is more dense (or heavier if you like) than the water. So the same thing happens with the rock in the mantle.
The cooler rock is heavier or denser and so it sinks beneath the lighter, hotter rock. Now the hotter rock has nowhere to go, which means it must rise. So that's how the convection currents form.
