How do spaceships fly in space?

They "fly" by their sheer momentum provided by reaction to bursts of rocket emissions of any length from short pulses (usually to correct the path) or longer bursts for major stunts like taking off, or slowing down to land on the moon etc. The momentum provided by a rocket engine "burn" happens because you have a fierce generation of expanding gases which press on a wall of the burn chamber, but are nut balanced at the opposite side of the chamber where they rush out unchecked.