This is all about gravitational maneuvers.
They allow to obtain huge accelerations/deceleration/velocity changes almost without using any energy. More heavy moons - more opportunities for maneuvers.
General idea is that if satellite trajectory at some point goes near heavy body (one of moons), by very small early adjustments from long distance before descending, it is possible to choose at which side of body to pass by and how close to it to descend.
Closer to moon -> bigger effect of gravitational maneuver -> bigger amount of free velocity correction (=energy) extracted from maneuver.
By the way, to use this, it is necessary to have entrance hyperbolic trajectory to Saturn/Jupiter systems which will be near as much as possible moons.
If there were no moons at all, trajectory would be always hyperbolic, and satellite will always leave gravitational field of Sat/Jup after some time.
If on some point of this trajectory to pass by heavy moon, it is possible to use gravitation maneuver for deceleration and to stay in grav field of Sat/Jup just for free.