Danforth requests this meeting between John and Elizabeth for selfish reasons. Danforth realizes that John has influence over many of the townspeople. He hopes that two things will happen when John sees his wife: that John will be so overwhelmed by seeing his wife for the first time since learning she is pregnant, he will confess to witchcraft in order to be with her; and that Elizabeth will, in her fragile feminine state, beg her husband to confess so they can once again be together. Paradoxically, Danforth underestimates Elizabeth's religious convictions; her faith will not allow her to encourage John to admit to such blasphemy, nor will she confess herself. John considers confessing, attempting to justify this option by saying that he is not fit to be compared with those martyrs who have hanged for crimes they did not commit. Of course, John realizes that he can not confess and retain his "name," or the upstanding reputation he has worked hard to establish.
If Danforth can somehow coerce John's confession, Danforth's court and the convictions he has already laid down will be justified. In the eyes of those villagers who respect John Proctor, if he were to confess to witchcraft, then perhaps it would not be so far-fetched to believe that others are witches as well. If those accused continue to deny their involvement in witchcraft, Danforth realizes that at some point, the villagers will question his decisions, and in turn, his authority.
By the time that act four rolls around in the play, hundreds of people are in the jails, the townspeople are starting to turn against the judges and the courts, a nearby town of Andover overthrew the courts and freed all of the "witches," and the judges and Parris are worried that people are going to start a riot. Parris even found a knife stuck into his door. Plus, the main accuser, Abigail, has bailed, hopping on a boat out of Salem--this seems to indicate that the townspeople are angry and upset at all of the charges, and will soon turn on the accusers and the courts.
Because of this atmosphere, Parris thinks that if they can get some prominent people of the town to confess, it might influence others to confess, and less people will die as a result. When you confess to witchcraft, you don't hang; they think that if they can kill fewer people and send them home, it might appease the feelings of the townspeople and tamp down on an insurrection. Their thoughts turn to Proctor--he is well-respected in the town. If he confesses, maybe other people will too, following his respected example. Then, if they confess, they'll be set free, and the townspeople will be appeased.
So the plan is to get Proctor to confess, to try to influence others to do the same, to calm the anger of the town. However, Proctor is remaining firm. They think that maybe if they can get Elizabeth to convince him to confess, he'll listen to her--because so far, he isn't listening to anyone. So, in the hope that Elizabeth will convince John to confess, Danforth lets them speak together. And, it does work, at first; however, in the end, John holds his integrity.
Does that help clear things up a bit? I hope so. Good luck!