Why does heating the air inside a hot-air balloon cause the balloon to rise up into the cooler surrounding atmosphere?

Heating causes the air molecules to expand outward from the bottom opening, making the air inside less dense than the outside air.

Explanation

When the air inside the balloon is heated, the molecules speed up and spread out, forcing some air out of the open bottom of the balloon envelope. This makes the inside air less dense than the cooler surrounding air, so the total weight of the balloon system becomes less than the buoyant force of the displaced outside air.