When air warms up, its molecules get further apart. So imagine two boxes, the same size. One filled with warm air and the other filled with cold air.
The one with the warm air (heat) in it will have less molecules, or less air in it. Conversely, the box with cold air in it has more molecules inside it because they can be packed tighter.
With less air inside it, the warm box doesn't weigh as much as the cold one, and so it rises.
In short... Warm air, or heat, is less dense and lighter than colder air, so it floats up like a balloon with helium inside it.