How is it that Jupiter rotates so fast on its axis and that the gas of this gas planet stays in place?

Answer

The gas stays with Jupiter for the same reason that our atmosphere stays with us. Because the acceleration of that gas toward Jupiter is greater than its acceleration away.

Jupiter’s rotational period is only two times shorter than that with which the Earth rotates on its axis; the rotational acceleration outwards is all together 60 times greater than with us (the effect of the faster rotation plays with the second power, and the acceleration is also proportional to the radius). But the attraction is also greater: the mass is 330 times as large as ours, the radius about 10 times, so the attraction is three times greater in total. All in all, the rotational effect at Jupiter is therefore relatively greater than at us, but the gravitational force of the planet remains stronger than the force outwards.

You also ask why Jupiter rotates faster than Earth. The correct question is why the Earth spins slower than Jupiter. The answer is the tides. The tidal forces cause a satellite (a planet of a star, or a moon of a planet) to rotate more slowly. Because Jupiter is farther from the Sun than we are, the Sun’s tidal forces have had less of a slowing effect on Jupiter’s rotation than ours.