It's at Jupiter's distance and beyond that ices were able to form out of the disk of material surrounding the early sun. Go much further in and there is too much energy from the sun for them to stay as solids (and will sublimate into gases); this is why asteroids are principally rocks and metals. So at this distance more of the materials of the planetary disk can form the base planetesimals.
At this point, your question is answered largely by a geometric consideration (or two) and one of Kepler's laws.
First, the geometric consideration. A circle of radius $r$ has area $pi r^2$. The bigger the radius, the more area. The material for Jupiter (or any other planet) came from an annulus: stuff outside one circle, but inside a slightly bigger circle. This annulus had a lot more area for the out planets than the inner planets, and so could contain a lot more mass.
Of course, that could makes us think that Jupiter shouldn't be the largest of the gas giants: it's the closest of them all to the sun, after all. The density of the disk needn't have been approximately constant throughout these regions, though. Quite possibly the density was such that Jupiter's region had more mass than the areas for the other planets. As HDE's answer (posted as I was finishing this) points out, these ices probably also helped stop materials from passing into the inner solar system, maintaining a higher density than you might otherwise expect in the inner solar system, as well as causing materials to sort of "dam up" right around Jupiter's orbit.
Now for the Kepler's law. The further you get from the sun, the slower your orbital period. Picking the correct units, we have $P^2=a^3$, where $P$ is the period measured in years, and $a$ is the semi-major axis of the orbit measured in AU. The further out you go, the slower you go around the sun; indeed, it's not just that it takes you a longer total time, but your actual velocity goes down. We can also see this as a Newton's law of gravity consequence. At Jupiter's furthest point from the Sun, the escape velocity is a little more than 18 km/s. At Saturn's maximum distance, the escape velocity drops to around 13.25 km/s. So things can go roughly 35% faster within Jupiter's orbit than they can closer to Saturn's, and they have less distance to travel to make a complete orbit.
What this means is that it takes longer for planetesimals to get close enough to each other to accrete together the further out you go, and there is a longer mean time between collisions.
Now, eventually, the Sun "turned on" and started blasting space with it's solar wind (before that, the heat came primarily from thermal radiation from the gravitational contraction of the sun). This ended up clearing out most of the unaccreted particles out of the solar system, stopping planetary growth (and removed portions of existing atmospheres; a very young Earth probably had a lot of H and He in its atmosphere, until the sun hit it with enough energy to knock any of it not locked up in rocks away).
So Jupiter was probably in a bit of a goldilocks situation. The average density of the region it formed in was probably higher than where the other giants formed in, it was at the perfect spot for lots of materials to start accreting early, and the accretion process would have been faster. So Jupiter is growing faster, and this gives it a competitive advantage: the bigger the growing planetesimals get the further their influence extends and the faster they can pull in more materials, and subsequently interfere with the growth of other planets (or planetesimals). Somewhere around a mass of 10-15 earth masses, the giants can start pulling in large quantities of the hydrogen and helium gasses. And, again, Jupiter likely hit this mass well before the other giants, and had more material to pull from, so it became much larger than the others could before the solar wind stopped the process.
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