In a hot air balloon, a gas burner is used to heat air to a temperature of about 212°F (100°C).
Since hot air is lighter and less dense than the cool air around the balloon, the heated air causes the whole balloon to rise. When the air inside the balloon cools down, or when the hot air is let out, the balloon goes down.
Everyone else simply states that warm air rises, but your question is Why? The answer is Gravity. Since warm air is less dense than cool air, cool air is pulled downward more strongly, forcing the warm air to rise. The only force at work is gravity.