How do space probes use other planets to propel them faster?

It doesn't "do a few loops" around the planet, as that would require LOSING some speed to go into orbit. It's a one time p-by. It's not because leaving a planet's gravity loses less speed than the speed gained by falling toward it. It's because the probe robs a small amount of the planet's kinetic energy as it moves along its orbit around the Sun. That energy is transferred to the probe. The planet's gravity is merely the temporary "tow line" allowing that transfer to happen.