Why do rockets fly in space?

The intuitive idea that an engine "pushes off" the air is false. A rocket works by expelling mass: it burns propellants in the combustion chamber, accelerates the resulting gases through a nozzle, and ejects them backward at several kilometers per second. Newton's third law states that this action produces an equal and opposite reaction on the vehicle. Nothing external is involved; the thrust is generated entirely inside the engine, the same on the launch pad as at 400 km altitude.

An aircraft jet engine draws oxygen from the atmosphere to burn kerosene, which is why it stops working as soon as air becomes scarce. A rocket avoids that limit by carrying its own oxidizer, whether liquid oxygen, nitrogen tetroxide, or the perchlorate used in a solid propellant. It's an expensive solution in terms of mass, since the oxidizer usually weighs more than the fuel, but it's the only one that allows an engine to ignite in a vacuum or on the surface of the Moon.

Contrary to popular belief, space actually improves a rocket engine's performance. In the atmosphere, outside pressure slows the expansion of gases in the nozzle and reduces thrust; in a vacuum that pressure disappears and the same amount of propellant produces more impulse. That's why upper stages use nozzles that are much longer and wider than those of the first stage, designed to work where there is no air. Moreover, without aerodynamic drag, a small push is enough to change orbit, and satellites can maintain their trajectory for years with tiny engines.