Why would someone who is already in pain wish to inflict it upon others?

Sometimes pain internally (mentally or other) and Physically (abuse or other) reaches a point where the solution isn't easy. You know you need something, but you don't know how to get it. Inflicting pain to oneself sometimes gives a sense of control in the middle of the storm. The hardest part is remembering storms pass.

Based on my experience, bullies usually get their motivation to inflict pain upon others when they're being bullied. Unfortunately, a lot of people think that bullying will make their reputation climb up in school. So when people are bullied, they try to join in, so that they won't be considered an easy target to pick on.

They want other people to feel the pain they are going through. Physically or emotionally. I guess you could say they want others to suffer like them.

I've been victim of second hand pain before. It's a main cause of why people are bullied at school or at work. People in pain tend to want others to feel how they have felt. If you yourself don't understand why you want to make others hurt, I'd like to tell you there's another way to cope. Another thing someone who's been bullied can do is try to not make anyone go through what you've gone through. This is my method for being in pain. Hope it helps

Well since they are in so much pain, they want to see it in others. They think it'll make them feel better knowing someone else is feeling pain. They might think that since more people are hurting, it'll make them better than others.

That must be because they are unwilling to face the truth and shift the blame onto someone else. However, it makes an endless cycle. If they hurt those close to them, they would get hurt in return. But it's possible for them not to stop because no one is guiding them towards the right path.

because they think if they suffer pain, others around that don't care to offer help, should feel the same emotional, physical pain

Sometimes, it is difficult for those in pain to deal with their emotions, so they feel as if they need to lash out instead of coping with their emotions

They're acting out instead of focusing on fixing their problems.

life has been tough for this person who is inflicting pain, life has not met their needs correctly, they're someone who wants to be heard, they want to let out some anger which is why they have no other choice other than to let it out. Don't take it personally :)