When you burn rocket fuel, it makes a bunch of hot gas. By pushing all that gas out of a (moveable) nozzle in one direction, it pushes the ship in the opposite direction.

ELI5:

It's like blowing up a balloon and then letting go of the neck. It flies off because every action has an opposite reaction. The balloon squeezes the gas out the neck, which pushes the balloon off in the other direction. This would happen in space with no outside air around the balloon too! We just use rockets instead of huge balloons because rocket fuel is a convenient way to bring up a large volume of solid or liquid fuel in a small space and then turn it into gas to shoot out the nozzle as needed.