Psyche represents a Sin Eater's power, which relates to bond one has with his Geist. This bond is in the sense of combined will, mortal determination and deathly purpose. A Sin Eater with a high Psyche level is more connected with the afterlife and less with the living world, while a low Psyche seem almost normal to the people around them. This is different from a Bound's Synergy, which is the balance of the bond. This connection as it gets stronger will allow greater amounts of plasm use.
It does have it's dangers. The stronger one's bond is, the more senstive the Sin Eater beomes to the energies of the dead. This means she needs to keep the connection strong by spending time with the dead, which in higher levels usually involves time in the Underworld. The high Psyche level also means one needs more anchors to the living world. The anchors have on a stronger affect as well, where disturbing the anchor point can disturb the Sin Eater the same way it would for a ghost's fetter.