From someone who was both a victim of bullying and a bully, I'll say this:
When I was in pain, even though I knew what I was doing was wrong, hurting someone else gave me a moment of 'control' that I didn't have. I was bullied at school, abused at home, and I'd tried many times to reach out to others, but after many, many failures thought that no one cared. Abuse was -familiar- to me. I knew abuse more than I understood anything else. I didn't -want- to hurt other people, but at the same time I truly believed no one loved me or cared. I figured it was better to be hated, then to get close to another and risk getting hurt again.
For a bully, It's a brief moment of feeling like you are on top of the world, and nothing can hurt you, and as someone who's been hurt many, many times... That feeling can be incredibly addictive, and you'll do anything to feel 'powerful'. I was insecure at myself, bitter and furious at the 'wrongs' I felt the world placed on me, and I wanted to show others the pain I felt. Is it logical? No, not at all. Was it right to hurt others for my own sake? Course not.
But when you are hurting, being abused, alone and in pain, you'll take almost -anything- to kill the pain. Some self harm, some drink their pain away or otherwise, and some abuse others. As for me? I wanted nothing more than to forget the pain in my own heart, and to feel like the world wasn't constantly crumbling under my feet. I didn't want to hurt anymore. I didn't want to cry. I wanted to feel like, even for a moment that I had -value-. I'm sure there's a lot of people out there that can relate to this. And despite how many bullies are often in denial, I can tell you a lot of them know what they are doing is wrong, and they know what it feels like to be in pain. They drown themselves in denial to avoid feeling even further pain from their cognitive dissonance.
It's a cruel world sometimes, and everyone chooses different ways to deal with it, healthy or not.
I'm not sure if -this- was the answer you're looking for, but hope this helps in giving you a glimpse into a former bully's head.