The question is interesting because there are a variety of fields of science through which we can look at bullying to try and understand it better. The scientific lens/approach I prefer is the behavioral approach.
Bullying is a behavior – so we can look to behavioral science to help us understand why bullying occurs, and how, potentially, to make it stop. It turns out that behavioral psychology – which is a well-established field of science, has a good understand of how and why operant behaviors (chosen behaviors as opposed to instinctual behaviors) are shaped, conditioned and habituated. This has to do with reinforcements and responses to stimuli in the environment. Sometimes those stimuli are other people.
If we look at bullying through this particular scientific lens (again, there are others that yield interesting insights as well), we see that both bully and victim are shaped by their choices and their actions, the actions of others and responses to these actions.
Why I like applying the behavioral science approach to bullying the most is because it really clarifies what is happening in a bullying dynamic and what we see is exactly what the science of behavioral psychology predicts we will see. Which is a fancy way of saying bullies bully because it works. The behavior is reinforced and rewarded. This reinforcement is unintentional, but it still happens as the people witnessing bullying or being victimized by bullying respond in a way that the bully likes. When victims and bystanders stop reinforcing the bullying behavior, the bully responds by cycling through an extinction burst, which manifests as retaliation or escalation of behavior, and again, this is predicted behavior.
The good news is that once we can see the behavioral dynamic playing out – we can also stop it using what behaviorists have learned about how to extinguish unwanted behavior. Animal trainers use these techniques all the time and again, the process of how animals, including human animals, unlearn behavior is so well studied that there is no debate about whether this works or not anymore. Basically, you have to stop rewarding the bully. The problem is that is incredibly hard to do AND you had to do this over time while the bully escalates through the extinction burst (which is part of the extinction process) until the behavior is extinguished. This isn’t easy to do because the escalation burst is so difficult to experience. But IF you can manage to do this and constantly not reward the bully over time, it will work and every study on the extinction process ever done has validated this process as being the most effective way to extinguish unwanted behavior.
The problem is that it is incredibly hard to do. Additionally, humans are free range animals which mean they can get their reward elsewhere, so it’s actually nearly impossible to retrain a bully to stop bullying entirely. About the best you can hope for is to train them to not bully a particular individual or in a particular situation, which you can control.
Again, applying the science of behavioral psychology isn’t the only science that can provide insight into the phenomenon of bullying, but it is a very good approach if your primary interest is in how to extinguish the unwanted behavior itself. What drives a bully to bully and the various reasons and social factors that may drive the onset of bullying behavior is better illuminated by other fields of science, like sociology, psychiatry and more.