It doesn't in 3.14. Prior to 3.14, all gems had either a mana cost or a mana reservation, and it was fundamentally impossible to have both. Thus Archmage had to be limited to skills with mana costs, because that was all the skills it could work on - skills with mana reservations couldn't have mana costs.
In 3.14 skills all have a cost, and some skills also have a reservation, neither of which have to be inherently of mana. As such Archmage can support any skill, because it's effect can apply to any skill.
EDIT: This is wrong - the cost-modifying part of Archmage could technically apply to any skill, but it's restricted to only support skills without a reservation (which is the closest thing to the actual meaning of it's previous restriction), because it's full effect only makes sense in the context of skills where the amount of damage you do has some relation to how many times you pay the cost.