Many people haven't dealt with domestic violence up close and personal. Sadly, it's so easy for people to ask, "Why doesn't s/he leave that (insert choice insulting word)?!" Unfortunately, it's really not that simple because:
-The abuse comes gradually. Many domestic violence survivors will tell you that their partner did not abuse them from day one. If they did, the victim would have simply left the person before it even started. Usually, the partner first appears warm, charming, intelligent, caring, maybe a bit insecure in a way that draws you in, and other positive attributes. After the person gets in the relationship, their abusive partner begins the first steps of abuse by trying to control and isolate them.
-Love is an emotion, not a rational thinking process. The abused partner has fallen in love with how the abuser initially portrayed him/herself. So when the abuser says something like, "I'm so sorry I hit you, babe. I promise not to do that again," his/her partner is more than willing to believe it. Unfortunately, instead of going back to being "good", the abuser is back at point one on the Cycle of Abuse, and the abused partner is left still hoping for the abusive partner to "realize his/her errors" and change. For a person or people trying to help the abuse victim, it can be very frustrating to witness a person not wanting to leave an abusive relationship at all. It becomes a sit and wait game for the abused to realize that they're in an unhealthy relationship while you are giving them help and support through their problems.
-The abuser makes the environment and situation of the relationship very difficult for the abused partner to get out of, or at least creates that perception of such. The abuser and abused may live in an area where the LGBT community is heavily looked down on and/or ignored, or may be a part of a religious group that does not agree with divorce whatsoever. Sadly, many domestic violence incidents go unreported. A huge part of the answer to the question of "Why does s/he stay?" is fear that if the abused leaves the abuser, his/her life will be ruined or worse, the abuser will kill him/her. Statistics out there show that most domestic violence related deaths occur when the abused person is leaving or has already left the abuser.