One of the primary aspects of a psychopath's interaction with others is that they relish playing upon others emotions in a predatory, victimizing way. They feel superior; they are not 'bound' by what they see as the weakness of being subject to, for example, sorrow, grief, loss and shock. This is to a large extent what I believe motivates serial killers. They know they are causing untold suffering and grief; They can't feel it themselves, yet they feel powerful knowing they can make others feel it, and in maybe also making them realize that it means nothing to them. It's self-flattery at the ultimate expense of others.
Arias' motivation in writing this letter was similar, imo. It was to re-live the killing, to give it life through the eyes of those who knew and cared for him most, to rub it in the face of those to whom it meant the most, and at the same time to confuse and confound them as to why and how someone would do it.
If you remember the first police interrogation with Flores, at one point she expressed interest in seeing the pictures out of "a morbid curiosity" ie, "I'd like to re-live the killing", but what really struck me is that when he left to get the pictures, when he came back the first thing she asks him out of the blue is "what kind of gun is that?" and then "I just bought a gun." Why steer the conversation in this direction? My guess is she either wanted to re-live the gun part of the killing, or frighten him with her cold ability to casually discuss guns in the midst of being accused of a brutal murder in which, in part, a gun was used. She was showing her detachment from the whole thing, saying in effect: "You don't know who you're dealing with, I'm not some scared little girl, I'll see your call, and I raise."
