Obviously, this will be too general: people stay for lots of reasons. I knew someone once who had a bad heroin habit, and while getting involved with a guy who beat her up if she tried to leave the house would not be my preferred method of detoxing, it worked for her. (She was still clean the last time I heard.) But generalizations might be better than nothing. I will also refer to abusers as ‘he’, and to their victims as ‘she’; this is accurate in the overwhelming majority of cases.
In some cases, understanding why someone stays is easy. A lot of women are afraid that their abuser would try to harm them if they leave. And with good reason: about a third of female homicide victims were killed by a spouse, lover, or ex-lover; and that’s not counting the women who are “merely” beaten, stalked, and so forth. Staying in a case like this, at least until you had figured out how to leave safely and cover your tracks, is not mysterious or perplexing.
Moreover, while I think the assumption that battered women stay because they are just dumb, or have staggeringly bad judgment, is wrong and insulting, there are a whole lot of battered women, and it would be very surprising if none of them stayed for such reasons. We asked women who came to our shelter when the abuse had started; one woman told me that her husband had thrown her from a moving car on their first date, at which point I wondered silently why on earth there had been a second date, let alone a subsequent marriage. But in my experience such women were a vanishingly small minority.