The mass and velocity (hence momentum) of the gas ejected from the rocket engine of the spacecraft causes the change in momentum of the spacecraft. The spacecraft moves off in opposite direction with increased momentum; but the total momentum of the system (spacecraft + ejected gas) is conserved.
It's Newton's Third Law, but another way of describing it is the conservation of momentum. Imagine sitting in an office chair with wheels on an ice rink. You have zero momentum.
Now light off a fire extinguisher. The gasses go in one direction, and to balance out the momentum you have to go in the other. A rocket works in the same way.
It shoots its fuel in one direction pushing it in the other. This requires a lot of fuel and is the reason why it takes a huge rocket to get a little capsule into space. There comes a point when all the fuel they can add does nothing more than carry its own weight so this puts a practical limit on maximum payloads.
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