One word explanation – it is got to be buoyancy.
Nearly all hot air driven balloons stay up in the atmosphere for exactly the same physical reason that that ships stay on the surface of the sea. A ship or boat stays buoyant for the simple reason that it is kept on the surface of the water by the the solidity of the water even thought it is a liquid. The force of pressure from beneath by the water equals the downward pressure created by the boat’s weight. So if a boat is too heavy it will sink! So the larger a ship is the greater the upward force is, which means the boat can support more weight. Basically if the ship, boat, seaplane, log or anything is less dense than the same volume of water it will float, and of course if the object is more dense than the equal amount of H2O it will sink. Also the heavier an object is than the same volume of water, the faster it will sink.
Hot-air balloons are buoyant because when air is heated it expands and for the same volume is less dense – the molecules become further apart. When this happens in a hot air balloon it floats because the air outside is thicker or more dense as it is colder.
Let us look at this explanation another way. The old adage that ‘heat rises’ is true enough but what is really meant by that phrase is that warmer air than the prevailing air temperature will rise. So when a bonfire burns you can see smoke rising in to the sky, you can see the hot rising air because there are particles of carbon etc contained in it that makes the action of the gases visible.
Have you ever played with Chinese lanterns? This a version of a hot air balloon. You hold a match or lighter underneath the paper lantern, the air inside heats up, expands, becomes lighter and it floats upwards because it is less dense than the cooler ambient air outside the chamber. A Chinese lantern is just a small hot air balloon.
If you want a larger balloon that will support more weight you just need a larger floating chamber – remember if the same volume of air is less dense than the surrounding ambient air, the chamber will rise. Is all about displacement, a larger floating ship will support a greater weight.
The size of a hot air balloon required to support a certain weight is just a displacement equation. For example the size of a balloon required to support an average man would be approximately thirteen feet in diameter and the air contained within would have to be one hundred and twenty degrees centigrade. Obviously with modern hot air balloons there are burner controls so the heat level can be adjusted, more heat the balloon rises, less heat the balloon descends!
Fora more detailed explanation go to the great Virgin ballooning website.