Puritans had a strong belief in the devil, and that Hell existed. Because of this, they felt that the devil could exert power and influence over people when they were on the earth; hence the belief in witches. They believed that witches were servants to the devil and did his work for him on earth. This belief is extremely evident in "The Crucible" as women and men are accused of being witches who have "signed his black book," meaning, signed their souls over to Satan. We see that Reverend Hale, who is a minister ordained to preach the word of God, also had an extensive knowledge of all of the "demonic arts" too; Satan was just as real as God was. Hale believed in Satan's influence so much that he was well studied in him and his various forms. Accusations of witchcraft only fly in a society where people believe in witches; Puritans did.
Another belief that the Puritans had was that the bible was not only the law of God, but the law of the land also. In Salem, they had a theocracy, which means that their government was also their religion; religious rules were the law. Because of this, witchcraft was punishable by law, because it was forbidden in their religion. Ministers became enforcers of the law. The bible was referred to for punishments and law setting. Pay attention to how often the townspeople refer to the bible as their guiding force. They refer to it for all of their dictates.
Another Puritan belief was that of being complete and total sinners, and that only baptism and strict adherence to gospel law would save people. In the play, Reverend Hale notes that people had noticed that John Proctor had not baptized all of his children--this put them in danger of not receiving salvation if they died. Hale encourages them to baptize their children immediately. Without baptism, a soul was condemned to Hell. Strict adherence to religious law was also needed to achieve salvation; Rebecca Nurse refused to confess to witchcraft because, she stated, "It is a lie! How may I damn myself?" She would rather die than tell a lie that might jeopardize her salvation.
Those are just a few thoughts, and I hope that they help; good luck!