Abigail Williams did not, in fact, bring witchcraft to Salem. What she had done, according to the play though, was to use the rumours of witchcraft in an attempt to take revenge on those she deemed her enemies and, secondly, try to win back John Proctor with whom she had had an adulterous affair.
The rumour began when Abigail and a number of girls were caught dancing in the woods at night by her uncle, the reverend Parris. The girls were dancing around a fire and Tituba, the Reverend's slave from Barbados, was leading the proceedings. When he jumped out at them, Betsy, his daughter, fell into a coma. Later, the daughter of another parishioner, Ruth Putnam, displayed similar symptoms. She would sleepwalk and cry that she wanted to fly.
Reverend Parris was obviously deeply concerned about the girls' actions and called for Reverend Hale of Beverley, an expert in the occult, to visit Salem and do an investigation. By this time, rumours of witchcraft had spread like wildfire through the village, so much so, that John Proctor, a farmer, commented that there was a pilgrimage to the Reverend's house.
The paranoid Reverend Parris feared that his position as parish priest was at risk if a finger should be pointed at him for not being mindful enough of his charges and allowing them to consort with the devil. However, visitations by other parishioners did not help much. The Putnams, for example, insisted that signs of witchcraft should be investigated by the soon-to-arrive Reverend Hale.
When the esteemed Reverend does come, he quickly discovers that Tituba had been the ringleader. He questions her and she, fearing to be hanged, confesses to the ignominious deed. On being urged, she mentions the names of villagers who she claimed asked her to harm Reverend Parris and to consort with the devil. Her confession opens up the sluice-gates and many other so-called confessions soon follow.
We learn, in Act Two, from the Proctor's maid, Mary Warren, that within eight days, thirty nine villagers had been arrested on charges of witchcraft. Mary has become 'an official of the court' and has to testify in court. She informs the Proctors that she had saved Elizabeth Proctor's life that day - she refused to provide testimony implicating her. She refuses to mention Elizabeth's accuser.
Elizabeth suspects that Abigail Williams, their previous maid, had accused her. She discovered that John was involved in an adulterous affair with Abigail and had summarily dismissed her. Elizabeth believes that Abigail has done this out of vengeance. She also believes that Abigail wants her out of the way so that she may reignite her affair with John.
It soon becomes clear that Abigail is leading the accusations of witchcraft against innocent citizens and that she has manipulated evidence against Elizabeth. As the chief instigator, Abigail is offered protection by the court. She and the other girls, to direct suspicion away from themselves, or at the urging of their parents (such as Ruth Putnam, who was told to implicate others so that her father may profit), or out of sheer wickedness or revenge, implicate many innocent villagers.
Their actions result in the tragic events which cause an uproar, confusion, fear and resentment in Salem and culminates in the execution of many innocents. A disaster indeed, for no prima facie evidence of witchcraft was ever discovered.