The diagram is in the rest frame of the planet. Now suppose a spacecraft is slowing down in the frame of the solar system. A planet is nearby, so it now starts accelerating due to its gravity and gains speed. Now, this speed increase is added to some component of the speed of the planet's motion when it comes out on the other side (this added component can be changed by changing the angle from which it approaches the planet, in order to maximize the slingshot effect). Once out of the planet's influence, the spacecraft has the same velocity as before, plus a component of the planet's motion, which allows it to travel farther out. This is the slingshot effect.
Trying to look at this in another way, consider the angular momentum of the spacecraft. As long as it is only under sun's gravitational influence, its angular momentum cannot change. However, once it's under the influence of another planet, the two angular momenta - one w.r.t. the sun and one w.r.t. the planet (due to their relative motion) - add, and once out of the gravitational influence of the planet, their relative components can be adjusted (based on the angle of approach towards the planet and the angle at which it flies away after the slingshot) in order to increase the angular momentum w.r.t. the sun, which in turn puts it in a larger orbit, allowing it to travel farther away than before.
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