Why does the planet Saturn have numerous (62) moons compared to the rest of the planets in the Solar System?
Saturn is a gas giant like Jupiter. It has everything from
tiny moonlets less than 1 kilometer across to the enormous Titan, which is larger than the planet Mercury. Saturn has 62 moons with confirmed orbits.
Why does Saturn have more moons?
According to http://solarsystem.nasa.gov/planets/jupiter Jupiter has 50 confirmed moons and 17 unconfirmed moons, whereas http://solarsystem.nasa.gov/planets/saturn Saturn has 53 known moons with 9 moons awaiting confirmation. So Saturn leads in confirmed moons, Jupiter leads in total moons including unconfirmed ones.
Saturn and Jupiter have many moons for quite a few reasons, one of the main ones being that they have an absolutely immense gravitational pull. During the early stages of the formation of our solar system, there would of been many planet-like objects floating around which our gas giants would have attracted. Furthermore, these planets are so far out in the solar system water would if frozen (which explains Saturn's rings of ice). Infact, we can show that the ice can form moons by looking at some of the moons of Uranus, some of them are half made of ice!
A few of the outer moons of our planets are captured asteroids. Phoebe, which is a moon of Saturn, is believed to have been a captured asteroid.
I haven't heard anything about Saturn having more moons than jupiter.
Considering your last statement, it's actually the other way round. Saturns rings evolve dynamically over time and must spread out beyond the Roche limit. They thus would deliver material to the outer region where it can coalesce into moons or at least accrete onto them.
Ah! Thank You for letting me know, I didn't presume that would be the case. I'll edit out the statement, seen as it's false.