Arthur Miller's play "The Crucible" depicts the hysteria which erupted in Salem, Massachusetts during the Salem Witch Trials. Although the play is not historically factual, specific elements from the trail significantly influenced the play.
As for the power which the trials gave to those who were powerless, the girls of Salem were initially the weakest (lead by Abigail Williams).
Historically, those who were deemed powerful were the one in charge of the Puritan lawmaking. Given that the Puritans defined Theocracy (divinely ruled government officials) as the ruling government. Therefore, the men of the government were those who were defined as the most powerful. It is not until the girls come forward making accusations of witchcraft that they begin to be given power.
The court believes that the girls know the identities of the witches of Salem. This knowledge gives the girls power over the village, the villagers, and the court. Regardless of the falsehoods told, the power is real and exists.