A far as the first series is concerned, the idea is brutal lifetime training combined with selective breeding.
As far as all the information in the books, about the same explanation. Strict breeding for physical and mental abilities. Only the fittest allowed to breed at all, only the perfect children allowed to live, harsh/brutal living conditions and training.
By the time Kellhus enters the scene, he's about 30 right? So presumably the peak of physical and mental ability.
There's some potential later on that Kellhus is using some kind of magics, akin to what we see Inrau use - but there's never any reference to that kind of thing. There's also the potential for some kind of special things only Kellhus can do that we see brief snippets of in TUC , but that extends far beyond the initial scope of the question I think.
So for Prince of Nothing, barring strange unexplained magic usage, the idea is that the 2000 years of Dunyain breeding and training gives them their abilities.
Also keep in mind that in the story, the Dunyain have trouble breeding with non-dunyain. This, I think, is an indication that their own breeding program has been so successful that the Dunyain are very nearly on the verge of being their own species (or just past that point). While 2000 years doesn't seem like enough time IRL, it seems pretty well established in Earwa that this is what has happened. (or at least this Could be a "reasonable" explanation within the books).
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