To understand "The Crucible" you must understand the Puritan mindset. It is hard to explain something so broad in one step here, but I will do my best.
Puritanism was a reaction to perceived corruption in the Roman Catholic church. The Roman Church's teaching on doing good deeds to earn salvation and purchasing forgiveness for sins with gold and silver caused a backlash that resulted in the Protestant Reformation. The Puritans were a part of this reaction.
The did not believe that you could earn your salvation, but they did believe that you actions were very good indicators of your spiritual condition. A big part of this was the willingness to "come back to God" when you strayed away. Keep that in mind.
When the Salem Witch Trials began, there were all kinds of accusations made by the teenaged girls at the center of all of the trouble. These accusations were taken very seriously by the Puritan leaders, who literally ran the town.
Characteristic of Puritan practice, those who were accused of these practices were given the opportunity to confess and "come back to God." As mentioned above, this was a very, very important part of their doctrine. If you were willing "come back", which means confessing that you've actually been away and want to return, you could be forgiven.
The accusations made were false, but some of the people accused went ahead and confessed to guilt and expressed a desire to "come back to God" just to please the ones by whom they were being judged. This gave the judges an out. They could say, "You were wrong, but now you are forgiven." Thus, pleasing everybody.
Others, as a matter of principle, would not confess to something they had not done. Rather they claimed love for God but did not make any effort to "come back to God" because they did not believe they had been away from Him. This was not satisfactory to the judges. They were in a position of proving guilt and forgiving it, not in admitting that the charges they had brought these people before them on were wrong.