All horse training methods that work, work for the same reason. As already mentioned they work on pressure release or negative reinforcement. They also work because of a related learning mechanism- classical conditioning which is simply the horse learning an association between an intitially irrelevent cue (such as a voice command or seat cue) and a consequence- usually a pressure cue (like a whip or leg/rein cue).
Looked at logically there is almost nothing that's inherently natural about most NH methods. Horses don't use bits, bridles, halters, saddles etc on each other. The most common way they sort out differences isn't by aggression but by avoidance- avoiding each other. Avoidance is often very subtle and its easy to miss the signs and only see the agression and think that's the main way horses solve their disputes. They certainly don't chase each other round and round and round, unless in play and they rarely voluntarily get in close to, approach or closely follow horses that are higher up the pecking order than them.
While many NH methods represent a massive improvement on the brutality of some of the "old" ways, they are no more inherrently humane or effective than the good conventional methods which don't call themselves natural.
Any training method or technique that uses a minimum of force, fear or fatigue to train the horse to reliably respond to cues will work because of negative reinforcement and classical conditioning, irrespective of the label that's given to it.