What is the exact reason why hot air rises up while cold air moves down?
As air is heated its temperature increases and this results in its volume becoming larger.
The same mass of hot air has a larger volume compared to that of cold air. Density is the mass divided by volume. As hot air has a lower density than cold air, the weight of cold air displaced by the same volume of hot air is higher than the weight of the hot air itself. The buoyant force on the hot air that acts upwards is larger than the gravitational force acting downwards. This makes hot air move upwards.
On the other hand, the weight of cold air is larger than the buoyant force that acts on it. The net force is a downward acting one that results in cold air moving downwards.
Small point add onto the great input by justaguide. The fundemental reason hotter air's volume increases is due to the fact that hotter particles vibrate faster. Thinking about a gas, this will have the end result of pushing them further apart, therefore increasing the space required by that gas. This will cause the expansion, reduction in density and everything as detailed above