For an ideal gas, the temperature(T), volume(V) and pressure(P) are related by the ideal gas law PV = n*R*T where n is the amount of substance in moles and R is the gas constant.
If pressure and amount of substance are constant, the volume is directly proportional to the temperature. As temperature increases so does the volume.
A hot air balloon is filled with air at a higher temperature than the surrounding air. The same mass of gas in the balloon occupies a larger volume as compared to the surrounding air. When an object is immersed in a gas, there is a buoyant force exerted on it which is equal to the weight of the gas that it displaces. The weight of the air displaced by a hot air balloon is greater than its own weight. This is the reason why it floats in sky. Though a hot air balloon floats, it does not continue to rise for ever. As it moves up, the pressure of the surrounding air decreases and this decreases the buoyant force on the balloon. It can only rise to a height where the buoyant force is equal to the weight of the balloon.