Well it goes without saying, but spoilers ahead
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Basically, soulbursting in the main body of the fluff is only told from Yvraines perspective, but it seems to act something similar to phoenix lords, or exarchs, when a nearby Aeldari is slain, she "inhales" their spirit, getting a new voice in her head, along with all their memories and emotion, and a burst of refreshing energy to knit together wounds. While she does muse if she could teach this to other followers of Ynnead, its never explicitly stated how it works for others, aside form her. The closest we get is the flavour text from the formations and psychic powers.
In the flavour text, its simply put that Aeldari draw on the swirling life energy being released by dying creatures in the battle, Aeldari or otherwise. For any Aeldari spirits they absorb, they add it to the collected bank of all previous souls they have absorbed, becoming like living infinity circuits. Doing so energises both them and the spirits they take in, both receiving a true "rebirth". For any other type of creature soul, they are merely an energy source, used and consumed for a burst of speed, quickened reaction times or what have you