The governing body of Salem, at the time of the witch trials, was a theocracy. A theocracy is where the law makers of the group (or established colony) are in power based upon their "divine guidance." What this means is that the Church made the rules and forced the rules to be followed.
The problem that exists within a Theocracy is the fact that a checks-and-balances system fails to exist. The laws created are made based upon religious ideologies and, therefore, looked at with a very different "eye."
The Theocratic government of Salem impacted the play given that it was the laws of the church the girls, and others accused, broke. For example, outside of Puritan law/belief, dancing was allowed. Likewise, many other laws the Puritans held were not necessarily consider unlawful in a Democratic government. Therefore, the Theocratic laws were what forced the hands of the governing body (the Church).
Really, there can be no law against witchcraft unless religion is playing a role in government. The Bible, Puritanism's core religious text, specifically states, "Thou shalt not suffer a witch to live" (Exodus 22:18). They interpreted this to mean, then, that God not only condones the punishment of witches but expects and demands it. In a community that is not a theocracy, such a law would not exist. Consider the law in 21st-century United States: the government is not allowed to tell us what to believe or how to practice our religion (although we obviously have disagreements about what this system should be like).
Furthermore, Salem wasn't just any old theocracy. It was a theocracy with something to prove. The Puritans left England because they felt that the Protestant Church had not done enough to distance itself from the Catholic Church. They wanted to purify the church (hence their name) and, in doing so, show to the world just how a perfect community -- without any separation between church and state -- should be run. The Puritans believed that they were on God's errand, and that if they failed in that errand, they would do so quite publicly and bring down God's wrath upon them. Therefore, their need to tightly control Biblical interpretation and the behaviors of those in the community tended to overshadow what was, originally, a theoretically righteous message. Salem, in this sense, was a crucible (as Miller points out with the title to his play): people were anxious to do what they believed God wanted them to do, and this tension creates the perfect breeding ground for hysteria. Thus, the original intention of the community, established, from the start, as a theocracy, contributes to the outcome of the play (and the trials).