Art mentioned Newton's 3rd Law of motion, which is correct, but didn't quite go far enough with the explanation.
Simply put, Newton's 3rd law is "For every action, there is an equal and opposite reaction." In rocketry, it makes no difference if there is something to push against or not. The exhaust of burning fuel is a chemical action. For that, there must be an opposite reaction, which in this case is thrust.
Actually, it takes less thrust in space to move a rocket than it does on Earth, since the gravity is much less the higher you get. That's why maneuvering rockets on spacecraft are so small. It only takes a small, short blast to turn or position an object. In blasting off from earth, a rocket must create tremendous thrust in order to lift off the ground and reach escape velocity. The rocket isn't really pushing against anything except gravity. It's just that the thrust created is far greater than the weight of the rocket itself.
It's much the same way that an balloon will move forward once you release it. Although it is only air coming out of the balloon, the volume of air is creating thrust against the opening in the balloon which is greater than the weight of the balloon, therefore propelling it forward.
@SpaceProg--yes, we have a long way to go in rocket technology before humans can go very far away from Earth.