Abigail's motivation for crying out witchcraft in the beginning is to twofold. First, Abigail and the girls fear that they are going to be severely punished (whipped) for their behavior in the woods of conjuring spirits with Tituba, dancing naked, casting spells, and drinking chicken blood; all of which are highly forbidden in the Salem community. So, when Abigail witnesses the treatment of Tituba (the whipping until she confesses she saw the devil) from Rev. Hale and the others, she immediately sees her way out of punishment. She claims, "I want to open myself. I saw Goody Good with the devil..." etc. In doing this, she realizes that Rev. Hale and the other authoritative people such as Rev. Parris and the Putnams, believe her, and so, she is free from what would've been a severe punishment. She is now looked upon like a saint. At one point, Elizabeth says, "Where Abigail walks, the crowd parts like the sea for Israel."
Second, now that Abigail has gained a sense of respect and power, she continues naming names, especially Elizabeth Proctor. Her motive here is pure revenge. She wants to remove Elizabeth Proctor so that she may have John all to herself. Elizabeth says to John, "There is a promise made in any bed." And Abigail is determined to keep that "promise" with John. The only way to do this, in her mind, is to get Elizabeth out of the way. Let's not forget also that Abigail says that Elizabeth has "blackened her name in the village."