This is only really relevant to cases where the loads are higher than the weight of the arch/chain (manmade arches typically are just symmetrical as the loads are far less than the arch's).
I'm sure someone here is capable of elegantly describing these loads in the language of math to describe the same conclusion. That's not really the why part, just a different language for explaining it.
In a hanging chain, the sum of the forces pulling down (due to gravity/load) and away from the end (due to tension) on a given link must point in a direction exactly opposite the angle of the next chain link toward the end. Else that chain link would rotate until the above is true.
In a stone arch, the sum of the forces pushing down (due to gravity/load) and toward the base (due to compression) on a given stone must point in a direction exactly equal to the next stone toward the base. Else that stone would rotate and the arch would break.