Not everyone on Siaral becomes a Breeder, but this is what the race was called by the fae, and they don't see a reason to change it. They are a humanoid race, created purely to look after the creatures they share a world with, so they weren't originally given any magical powers. On this planet the fae planned on the magic of the creatures being collected rather than the people. Now many of the Breeders do have one of three forms of magic that have become normal on this world, in part due to the magical volcano.
The first form of magic is wish magic. Some Breeders can grant the wishes of anyone on any world, including Taithmarin, Kankirin, and Aerith. Normally they are careful about whose wishes they grant, but occasionally they make mistakes and grant wishes that adversely affect other people, and they don't often grant wishes for the anyone on the lost worlds. It's possible that the Breeders with wish magic know more about the worlds of the web than the walkers.
The second form is knot magic. Using different types of knots these Breeders can tie spells into cords for later use or to sell on to people who don't have this ability. Often these Breeders travel to other worlds to learn spells from different races to include within the knots, as they can learn almost any type of magic known in the web. Some people think that these Breeders have the most terrifying ability as they can learn all the types of magic and the walkers watch over them more closely than any of the other breeders.
The third form is scroll magic. Breeders with this ability can create magical scrolls that can be used by almost anyone on the web. Most other written magic can only be used by races with magic, but these scrolls can be used by races without magic too. They are more expensive than any other written magic, but people are happy to pay the money in order to use magic they may never have been able to use.
Each type of Breeder wears a band around their arm to tell people which type they are. Those Breeders who don't have any of the Breeder specific magics or want to raise the magical creatures of Siaral often travel to one of the other worlds.