There are several reasons that Elizabeth feels, and accurately so, that Abby wants to kill her. One reason that Abby wants to cause harm to Elizabeth is spite. Elizabeth, once she suspected the affair between Abby and John, fires her and kicks her out of the house. This was damaging to Abby's reputation; when a girl is fired, the word spreads through town, and rumors started. Because of those rumors, Abby couldn't find employment anywhere else, and had to go live at her uncle's house, the Reverend Parris. So, all of this makes Abby pretty mad. She even asks her uncle, "Do you begrudge my bed, Uncle," meaning, do you regret taking me in? Then she goes on to call Elizabeth a "snivelling, gossiping liar" who was spreading lies about her to keep her from getting employment. So, her anger was so intense that spite was a definite factor in her getting Elizabeth arrested.
The second reason that Abigail would want Elizabeth gone is simple jealousy. Abby is in love with John; this is very apparent when John visits the Parris household and Abby throws herself all over him and tells him that she thinks about him every night. She loves John, and wants to be with him; Elizabeth IS with him, so she is jealous. That jealousy and envy drive her to try to get rid of Elizabeth and take her place.
The last reason, that isn't quite as obvious as the first two, is a sense of twisted self-righteousness. In an optional act that is often left out of the play, Miller had Abby and John meet in the woods. John went to try to talk Abby out of her accusation against his wife. She refused, and insisted that Elizabeth was a hypocrite, and that "the world will be white again" when the likes of Elizabeth--who, according to Abby, professes being a kind, Christian woman but is really a mean, horrible person--are eradicated from the town. Abby said that it was she who was truly deserving of John in marriage, and that it was her personal mission to condemn all of the hypocrites in the town. So, she has a rather bizarre and twisted sense of self-righteousness, like she had been called by God to get rid of Elizabeth. Whether she really believed that or it was just an act, we don't know, but that is what she expresses in that act.
I hope that those thoughts helped; good luck!
Elizabeth thinks that Abigail wants to kill her for John Proctor. John&Abigail had an affair & Abigail is still trying to get with him. So Elizabeth believes that she is going to kill her so that Abigail can take her place as his wife.