Let's imagine that the air molecules in the balloon are M-U-C-H bigger than they really are so that we can see what's going on.
When the gas burner at the base of the balloon is turned on, the air molecules in the balloon start to heat up and move around more.
The air molecules have moved away from each other so much that some of them are forced to come out of the bottom of the balloon.
Fewer and fewer molecules are left in the balloon. It starts to rise because the total mass of air in the balloon has decreased, but the volume has remained the same, so the balloon now has a lower density than the air around it.
Lighter Then Air Lifting Gases
Hydrogen
Cheaper to produce then helium, but more dangerous (Hindenburg).