The aspect of the Puritan society that allowed the Salem Witch Trials to happen was the fact that they believed that they were "depraved sinners" and lived in a constant state of fear of going to hell. In addition, actions meant little, but each person needed to undergo "a radical undertaking that demanded each individual to plumb the very depths of his heart and soul."
To think what this means in action, think about living in the Massachusetts Bay Colony and working as a farmer. Perhaps it was cold during a week of the growing season and the corn didn't grow the way it should. This could be a sign that this farmer was out of God's favor and would probably suffer eternal damnation. This farmer better do some deep inner searching and figure out what's wrong with him on the inside in hopes that God will have mercy on his soul.
This attitude leads to the desire to root out all temptation, including anything that might be different, which can also lead to temptation.
This leads to the witch hunts.
Salem was struggling under several controversies before the witch trials took place. Salemites argued over land and animals and property. This may have led someone to make the false claim of witchcraft on an enemy, which led to this person accusing another of witchcraft, and so on.
This climate of fear from God and fear of neighbor created a perfect environment for the mass hysteria that occurred.