How does a rocket work in empty space, with nothing to push against?

A rocket pushes against its own exhaust, not against the air or the ground. The engine burns propellant and throws the hot gas out of the back at enormous speed, and by Newton's third law the gas shoves the rocket forward with an equal and opposite force. Nothing outside the rocket is needed, so it keeps working in the vacuum of space. In fact, rocket engines work more efficiently outside the atmosphere than inside it.

Picture standing on a skateboard and hurling a heavy ball backward: you roll forward. A rocket does the same thing, only with a stream of gas leaving far faster than any arm could throw. Rocket exhaust typically leaves at about 2,500 to 4,500 metres per second, which is the right order of size for reaching the speeds that orbit demands.

The part that fooled people for a long time is that a jet engine needs air to burn its fuel, and a propeller needs air to bite into. A rocket carries its own oxidizer, the ingredient that lets fuel burn, so it needs no air at all.

In 1920 the New York Times ran an unsigned editorial that scoffed at the American physicist Robert Goddard for suggesting rockets could work beyond the atmosphere. It argued that after leaving the air a rocket would have nothing to act on. Goddard had already run an experiment showing a rocket performs in a vacuum, but many scientists were not yet convinced. On 17 July 1969, the day after Apollo 11 left for the Moon, the paper printed a short item headlined "A Correction" admitting the error.