"In orbital mechanics and aerospace engineering, a gravitational slingshot , gravity assist maneuver , or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the path and [speed](Speed - Wikipedia) of a spacecraft, typically to save propellant and reduce expense. "
— Wikipedia.
It is actually easy to understand the principle. Imagine you, standing on platform, throw a ball at 10m/s at a moving train at 20 meter/s. The driver will see the ball traveling towards him at 30m/s and leaves at 30m/s, while you will see the ball traveling at 50m/s toward you because the train has a motion of 20 m/s.
The ball is the Earth, the train is Jupiter, and the platform is the Sun. Earth travels toward Jupiter and gains the speed by having a “gravitational collision” with it and gains the kinetic energy from Jupiter while Jupiter loses the same amount of energy. Earth loses kinetic energy if it goes against the velocity direction of Jupiter, which you can imagine as throwing a ball at the back of the train in the previous analogy. This also means gravitational slingshots can alter the speed of objects up or down.
The heliocentric velocity(velocity in the reference of the Sun) change of Jupiter due to the loss of energy can be ignored because of its huge mass compared to that of any spacecraft, but not if it is Earth being the “spacecraft”. Earth has 5.97×10^24 kg of mass, while Jupiter has 1.90x10^27 kg of mass, so Jupiter’s heliocentric will decrease by about 0.25%.
In fact, NASA has done many missions applied with this principle. For example, the Voyagers, Kepler, etc.
[Plot of Voyager 2’s heliocentric velocity against its distance from the Sun, illustrating the use of gravity assist to accelerate the spacecraft by Jupiter, Saturn and Uranus. To observe Triton, Voyager 2 passed over Neptune’s north pole resulting in an acceleration out of the plane of the ecliptic and reduced velocity away from the Sun.]