Hot air balloons are not like the balloons we commonly see. Instead of having stretchy, elastic walls, hot air balloons are made of air-tight (or nearly so) fabric.
In addition, hot air balloons are not filled by adding air, instead a fire is lit under the small amount of air already contained in the balloon.
As the temperature rises, so does the volume (Charles' Law).
Since the mass is the same (no air is added) but the volume goes up, the result is a lower density. As a result of this new, lower density, the air in the balloon (and the balloon and basket with it) will float in air.
To make the balloon go higher, you need only warm the air more, lowering the density further. To go down, you allow the air to cool, shrink and become more dense.
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