Giant-planet moons
Why does Saturn have so many moons?
Gravity’s reach grows with mass. Jupiter and Saturn are the two heaviest planets, so a huge volume of space around each one is a place where a small body stays with the planet instead of orbiting the Sun on its own. Some of those moons formed there. Many more were captured. See Jupiter’s moons and Saturn’s moons.
One giant, then a crowd
Titan is the moon that formed beside Saturn the way planets formed beside the Sun, and it holds almost all the mass that orbits the planet — a world bigger than Mercury, with weather and rivers of methane. After Titan come a handful of mid-sized icy moons, and then the census dissolves into hundreds of captured ice-rocks a few kilometres across, on long, tilted, often backwards orbits.
That is why the count keeps changing: the big moons were all found long ago, and what survey telescopes keep adding is the small captured crowd at the bottom of the list. “How many moons does Saturn have” is a census question, not a fixed fact — the honest answer always carries a date.
The rings are not failed moons on the list
Inside a certain distance of Saturn, its gravity stretches an orbiting body harder than the body’s own gravity can hold it together — so close in, there can be rings but no large moon. Saturn’s rings live inside that line. Right at its edge sit tiny shepherd moons, herding the ring edges sharp; they are on the moon list, while the countless ring particles they herd are not.
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Why the rocky planets keep almost none
Earth has one large moon from a giant impact. Mars has two tiny ones. Mercury and Venus have none. Close to the Sun the region where a planet’s gravity beats the Sun’s is small, so there is little room to keep a captured body, and less leftover ice to build one. Mass and distance both decide the guest list — Saturn is heavy and far, so its list is the longest. Jupiter’s answer is the same physics with even more mass behind it.