Well... it's similar to anxiety disorders or... or I suppose you could think of it as a form of torture. When something is causing you pain, sooner or later you'll do almost anything to make it stop. That's a motivator for a lot of serial killers, regardless of diagnosis — they have a drive to do something that will either alleviate their mental pain or bring them pleasure they can't find elsewhere. It's a basic human reaction that's been twisted by whatever delusions they're under.
Schizophrenia is sensitive to this kind of escalation, since it can and often will involve auditory hallucinations. If you're under constant emotional and psychological stress and a voice tells you it'll all go away if you just do this one thing... it's not difficult to imagine that the possibility of relief will eventually be enough to drive you to do just about anything, whether it's kill someone else or kill yourself.
Like I said earlier, schizophrenia itself does not cause aggression but the living situation of those living with the disorder can very well lead to that. Self-medicating with alcohol and drugs can worsen the symptoms and unemployment and homelessness is common. Add that to the misconceptions about the disorder itself and you soon have perfectly good people being forced to do unspeakable acts because they can't get the help they need.