A rocket lifts a weight in the same way as an inflated balloon shoots off when you let the air escape from the nozzle.
The air sprouting out of the balloon thrusts it forwards. This is actually down to Newton’s Third Law of Motion which states in simple terms that for every action there is an equal and opposite reaction. With a rocket, the principle is the same. What comes out of the rocket’s nozzle is essentially very, very hot air.
The fuel in the rocket is ignited and as it burns it shoots out of the nozzle, thrusting the rocket and whatever weight (or payload) it has to carry with it.