These extra impacts increase the individual pressure of the gas molecules, and as their pressure rises they begin to move apart. Expanding allows the pressure of the hot air to drop until it equals that of the cold air around it. This is what happens inside a hot air balloon. The air molecules take up more space because each molecule has more individual pressure. This means there are actually less air molecules in the balloon than there is in an equivalent volume of cold air around the balloon, so the air in the balloon is lighter.
Light things float on heavier things which means the balloon rises up.
Gases follow exact laws as to how much space they occupy at different temperatures, so we are able to use these to calculate exactly how much hot air we need in a balloon to lift it. A litre of air weighs about 1 gram, but if you heat this to 34 degrees Celsius it weighs only three quarters of a gram. So one litre of air heated to that temperature can lift 0.75 grams. You need 600,000 litres of hot air to lift 450 kilos of balloon and basket. That's a lot of hot air for something that can only go where the wind blows.