In The Crucible why does Elizabeth think that Abigail wants to kill her?

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!