Giles Corey is desperate to convince the court that "Thomas Putnam is reaching out for land" and that this is the reason why his daughter continues to accuse people of witchcraft. He has come to the court with John Proctor and Francis Nurse in order to present evidence of their wives' innocence and to accuse Putnam of wrongdoing.
Giles breaks down in front of Deputy Governor Danforth because his wife, Martha Corey, has been accused of witchcraft and is now being tried for this crime. He says that the girls who are doing the accusing are "tellin' lies about [his] wife." Early on in the play, Giles told Mr. Hale that his wife often reads strange books and that he cannot pray while she reads; however, he said, when she puts the book down and leaves the house, he can suddenly pray again. This seems to have started a chain of events that led to an accusation against her and her eventual arrest and subsequent day in court. Now, knowing that she will never confess to something she did not do, he fears for her life. He cries and says that he never suspected or accused her of being a witch. Now, he feels that he has "broke charity with the woman."
In short, Giles breaks down in front of Danforth because he is fearful for his wife and feels guilty for what he said that must have led to her trial.