Basically it boils down to two things:
1) When a pact is made with the Haemonculi coven, a permanent portion of the Eldar's soul is left with the Haemonculi.
2) If the Eldar's will is strong enough some of their spirit can linger in bodily remains.
The regeneration process is based around pain. The Dark Eldar are rejuvenated by witnessing agony and pain. They literally drink in the dark energy that is present in Commorragh and the Hamonculi labs in particular. If saturated with enough of it it can heal any wound.
Heresy's Background FAQ. (Fluff Project)
I do love how we don't even need CotE to comment anymore, chances are he has already said something intelligent before that can be re-applied to the current situation.
CotE Reviews: Prospero Burns (HH Review)
, Age of Darkness (HH Review)