Abigail controls the other characters through threats and strong diction. Like many bullies, Abigail believes the words she is saying and is not willing to back down. We see this in Act I when Mary Warren and Betty are worried that they will get in trouble for their nocturnal actions. Abigail threatens them with harm by reminding the girls about her past and her ability to kill any of them because of it.
ABIGAIL:Shut it! Now shut it! Now look you. All of you. We danced. And Tituba conjured Ruth Putnam’s dead sisters. And that is all. And mark this—let either of you breathe a word, or the edge of a word about the other things, and I will come to you in the black of some terrible night and I will bring a pointy reckoning that will shudder you. And you know I can do it. I can make you wish you had never seen the sun go down!
In Act III, as her power grows, she turns towards Danforth when he questions her actions. She threatens him asking if he thinks just because he is a minister does he think the devil cannot get to him. Her threat seems clear enough- he should watch out before she calls him out for witchcraft too.
Let you beware, Mister Danforth—think you to be so mighty that the power of Hell may not turn your wits?!—beware of it!
The other reason Abigail gets away with what she does, is that her claims are so outlandish that no one can imagine that she would be lying. If she was, which we know she is, she would be planning to murder her neighbors. While we see her malicious nature, this is not normal in the Puritan culture, and so they are unprepared to deal with her.