From the academic research I have done into this and from empirical evidence I can say the following.
They are motivated by fear and insecurity, and the need for self-preservation in a world about which they are fearful and insecure to an extent beyond that of those with a normally developed emotional intelligence.
In relationships, they have an over-weaning need to exercise power and control, often by any means, since self-preservation is of paramount importance.
The control takes the form of the carrot and the stick. The carrot takes many forms masquerading as generosity and charm, but is calculated to keep the subject subjected and 'on-side'. The stick also takes many forms, like withdrawal and/or reversal of the carrots, and, crucially to get other subjects to act in concert with them. For example, by contriving some injurious fault of the subject and discussing this 'injurious fault' with their other subjects, so that the subject is tried and convicted in absentia. Here is another facet of the bully, they are expert at playing the victim. The guilty blame the innocent to excuse their guilt. The first the subject gets to know about this is when they, for example have the normal social support withdrawn and possible some spiteful and anonymous retribution occurs. The subject never really gets to know what the causality is of these situations, since preserving the subject's ignorance is an important tool in preserving power.
Bullies are so insecure that even the slightest hint that a subject is not totally in their thrall will provoke many, if not all of the bully's arsenal of control.
Bullies need to 'divide and rule' by a constantly shifting sets of alliances and contrived enemies.
Bullying falls into two general types, the inveterate and the inadvertent. The inveterate is the dyed-in-the-wool bully who can be the head bully or a bully further down the bullying chain of command. The inadvertent go along with the bullying status quo for fear of receiving the same treatment.
It is this last bit that often motivates the inveterate to identify through their highly developed low cunning to identify a strong character (those who have their own minds) to victimise as an example to their subjects.
When a bully is in a situation where there exists a more powerful figure whose power is unassailable, then they contrive to be the power-behind-the-throne.
As alluded to above, the bully often accuses the victim of bullying.
How to deal with bullies
It is important for a victim to know it is not their fault and that they are not responsible for the bullying they suffer, especially since victims of bullying are often intelligent, competent people with a high moral sense.
Unfortunately, there seems to be only one way to deal with bullies and is a lot easier said than done, and requires courage, perseverance and stamina. Bullies need to be confronted relentlessly.
The victim who tires of their victim-hood should know that they are involved in a to-the-death struggle. This does not mean that the struggle will cause the death of an adversary.
Bullies will never be appeased. It is pointless to try.
When bullying accusations fly, fling real accusations back.*
When accusatory questions (questions keep the questioned on the back foot and cedes advantage to the questioner), don't answer, except with another question. You can be fair (although God knows why) and answer the question, but follow up with a question. Never answer immediately. Even if the pause is only a few seconds, it is effective, since it says that the victim is not spooked into a knee-jerk, fearful wish to appease the bully.
“I'll have to think about that one.” And, “I'll get back to you.” are good, especially if the victim goes on to ignore the bully.*
In discourse with bullies, be wary of questions or statements (as rhetorical questions) prefaced with things like, “So you're saying . . ?” And, “Don't you think that . . ?”
*Be prepared, be diligent in constant vigilance and take notes - lots of notes – bullies are well practised in their activity, victims have to learn the battle.
When the bullying has a large element of physical aggression, the victim must appeal to an equitable and canny higher power. This higher power/authority will need evidence, but bullies can be trapped into their activity in such a way as to have it observed and recorded. Physically to confront the bullies is dangerous.