Bigger is better.
Most moons, especially those of gas giants, are not "formed", they are just "captured" (unlike our Moon, which could have been captured, but probably was formed in a much more exciting way).
Jupiter is the most massive planet in the solar system. It stands to reason that it has a larger region of gravitational influence (where its influence outweighs the force due to the other planets and the sun). So, it's easy for it to capture rocky masses.
If you have a look at the contours on the following image (Ignore the Lagrange points marked on it, I only want the contours)
the circular area around the Earth is more or less the area (there's a velocity dependence here which I'm not getting into) in which a moon-like body can form a reasonably stable orbit. The size of the small "well" will increase as the planet moves farther from the sun, and also when the planet is more massive.
Jupiter is both pretty far away from the Sun, and is very massive. This leads to a huge sphere of influence.
The asteroid belt may have something to do with this too, but I doubt it (it's pretty far away). However, if we assume the "half-baked planet formation" theory for the formation of the belt, Jupiter may have leeched off much of the mass that would have otherwise become part of that planet during the formative period.
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