- Level 1: Helps Eggs hatch a little faster. The effect lasts three minutes. (1.25× as fast)
- Level 2: Helps Eggs hatch faster. The effect lasts three minutes. (1.5× as fast)
- Level 3: Helps Eggs hatch much faster. The effect lasts three minutes. (2× as fast)
In previous games at least, hatching is determined by egg cycles. Each egg has a number of cycles remaining, and when the step counter hits 255/256 (depending on generation), the cycle counter decreases (and the step counter resets to zero). When the cycle counter reaches zero, the egg hatches.
The step counter also resets to zero when a player does one of several egg-related actions (picking up an egg from the Daycare Man, etc). This means that it can take more than 255 steps to decrease the cycle counter, depending on the player's actions.
The abilities Flame Body and Magma Armor make hatching faster by decreasing the cycle twice instead of once when the step counter rolls over. This makes hatching twice as fast, for the most part. However, this only counts for complete cycles. It specifically does not mean that the egg cycle decreases 1 per 128 steps, for instance, since the step count can be reset. It also means that an egg with a single cycle remaining won't hatch faster than one with two remaining (which means that it's more effective for all even-cycled eggs).
This makes sense, and is easily done, since the only multiplier is x2 and you don't need anything but integer math.
However, with the new Hatching Power, you can have a 1.25x or 1.5x multiplier as well. I can't see how you could easily reduce the cycle count with those multipliers, unless egg cycles are now represented as something different than integers (1,2,3, etc). If the egg cycles are now counted with extra precision, that would be worth knowing.
Another alternative would be to decrease the number of steps that count as a cycle. For instance, instead of the egg cycle decreasing at 255 steps, it may decrease at 170 (255/1.5) for power level 2. This would be good to know, since (for example) it would mean you're less likely to reset the counter as far when picking up new eggs.
Is there any evidence for which method is being used? I can't find anything reliable that says anything beside "makes hatching faster". My gut tells me it's the second (since I swear I've had two eggs hatch less than 255 steps apart), but I've long since learned not to trust my gut on game mechanics and there's no handy Poketch app in this generation.
In case you're wondering, this is where my brain goes when I'm 900 Eevees into a Masuda run for that sweet blue Umbreon.
I've been watching my eggs a bit more closely since asking this question (still no shiny ><), and I believe it reduces the number of steps in the step counter (or adds extra to the counter for each step, same diff). My reason for this is the short time between hatching eggs in a batch. On several occasions two eggs have hatched in what "feels like" less than 255 steps.
For example, on Route 7 by the Day Care (Power level 3 + Flame Body), I had one hatch at the fork in the road (between the bridge and Camphrier), then another at the doors to the Chateau, after biking straight west. I don't have a step counter, but I know that in previous games, this distance would be less than 255 (about half that seems right). I don't know exactly what counts as a "step" in this game, though (due to fluid movement), so it's hard to be sure.