Alright, telu covered some of this already but I think I can explain it better.
1. If you run off the edge, there is a brief interval in which you are off the platform but the game still treats you as grounded (at least for the purposes of jumping). Or something like that.
2. If you jump in this interval, you jump immediately, skipping or drastically truncating the usual jumpsquat animation.
3. You can fast fall in this interval.
4. Fastfalling, then jumping in this interval produces the Akuma Jump.
5. It looks to me as if the Akuma Jump's odd height is a combination of two factors: first, fastfalling lowers the start point of the jump, and second, it seems to me that the momentum of fastfalling works against the jump (although I could be imagining it--normally fastfalling or not has no effect on the height of a double jump, but the game treats Akuma as a single jump so maybe it's different).
6. The timing is pretty darn strict. I don't know how to measure it, but it can't be more than 2-3 frames.
Also, there's a surprising amount of precision regarding horizontal momentum when using the technique. It is possible, for instance, for Sonic to Akuma Jump between the side Battlefield platforms--but only if executed perfectly (it's difficult even if in training mode and slowed down). Even the slightest hesitation in returning the control stick to the forward position results in him falling short. Mixup possibilities? Maybe.