The passages that you are looking for are going to be at the beginning of act four. In this act, some time has passed since Proctor was arrested. In fact, many hangings have occurred, and the townspeople are starting to question whether or not this entire trial has been a good idea or not. People are no longer cheering on the judges or Reverend Parris; in fact, the tide of public opinion is turning against them. Andover is a town not too far from Salem, and there are rumors going around that there isa
"rebellion in Andover...Andover have thrown out the court...and will have no part of witchcraft."
So, Andover has realized the error of the courts, and has thrown out the judges, released the accused witches, and won't hear any mentioning of witchcraft any more. So, Parris fears that
"there be a faction here, feeding on that news, and...I fear there will be riot"
in Salem as a result. The combination of Salem's already mistrusting public, in combination with the news of Andover's overthrowing of the courts, make the judges antsy. So, they feel if they can bring out Proctor, and "help" him to confess, he won't be hanged, and that will appease the people. They think that if he confesses, maybe some other people of "great weight" in the town will also confess, and the public will see that they aren't being hanged, and so won't rebel against the courts.
To sum up, Andover makes the judges fear that their courts will be overthrown, which makes them want Proctor to confess; they allow Elizabeth to speak to him, and in this poignant scene, they express forgiveness and remorse for past actions. This gives John the courage to face the gallows with a clean conscience, as a strong man at peace with himself.