I would say that the hysteria begins with the arrival of Mrs. Putnam. She and her husband have decided that witches must be responsible for the deaths of their little children, and she sends her daughter, Ruth, to Tituba, the Barbadian woman who is enslaved by Reverend Parris, to conjure the dead babies' spirits. When Ruth becomes suddenly ill the very next day, Mrs. Putnam jumps to the worst possible explanation: that a "sucking mouth [is] pullin' on her life too." In other words, Mrs. Putnam suspects that a witch must be afflicting Ruth now.
When she comes to see how Betty Parris is faring, as she is ill too, Mrs. Putnam reports that a neighbor "saw [Betty] [flying] over Ingersoll's barn, and come down light as a bird." Despite Betty's father's denial, Mrs. Putnam insists that it is sure that Betty did, indeed, fly. Her heightened emotional state leads her to suggest certain names to Tituba, as she believes that Sarah Good and Sarah Osburn are witches who are guilty for the deaths of at least a few of her children.
When Tituba becomes desperate to confess (because she's been threatened with a beating and a hanging if she does not), she says the same names that Mrs. Putnam has already suggested to her: Goodies Good and Osburn. People are most likely to believe those two women are witches because the prosperous and high-status Mrs. Putnam has already named them. The list of names grows from there when Abigail Williams discerns how she might use the hysteria to her advantage.