Gravity assist

Seen from the planet, the spacecraft falls into its gravity well, speeding up to a maximum at closest approach and slowing again on the way out. It leaves with the same speed relative to the planet as it arrived with — only its direction has changed, like a cyclist coasting down into a valley and climbing the far side.

The key is that the planet is itself moving around the Sun. When the spacecraft is timed to pass behind the planet in its orbit, then seen from the Sun the planet's orbital velocity is added to the spacecraft's, so the spacecraft departs carrying extra orbital angular momentum taken from the planet. In exchange, the planet loses some of its orbital energy: its orbital momentum decreases by a tiny amount, while the momentum the spacecraft gains is significant compared with what it already had.

A flyby can also remove energy, when the spacecraft passes in front of a body in its orbit: the Galileo spacecraft decreased its energy relative to Jupiter with a flyby in front of the moon Io, which reduced the propellant needed for Jupiter orbit insertion. Voyager 2 used the technique at Jupiter, Saturn and Uranus in turn; Cassini took two boosts from Venus, one from Earth and one from Jupiter to reach Saturn.