What does Proctor tell Hale about why the children are ill? How does he claim to know?
In Act II, Mr. Hale goes to John and Elizabeth Proctors' home to ask them some questions regarding their standing in the church and their religious habits. As he's about to leave, Elizabeth implores John to tell Hale what he knows about the girls' motives and actions. He says, in part, "I know the children's sickness had naught to do with witchcraft [....]. Mr. Parris discovered them sportin' in the woods. They were startled and took sick." In other words, he indirectly accuses the girls of lying, something they've been doing for the past two weeks or so, when they've made their accusations of witchcraft against others in the village. He suggests that the girls were simply shocked when Reverend Parris frightened them and that they've been making everything up since then (ostensibly to protect themselves from blame for what they were doing in the woods).
Proctor admits that his knowledge of this comes directly from Abigail Williams, and he claims that he's not said anything until now because he didn't realize that "the world is gone daft with this nonsense." However, as Elizabeth feared, it seems to be too late. Hale argues with John, claiming that it isn't nonsense because several others including Tituba and Goody Good have confessed themselves to be witches.
What John Proctor tells Reverend Hale is that it is not witchcraft that has made the children get sick. He implies that their sickness has to do with them being startled and wanting to get out of being in trouble.
He claims to know this because of what Abigail Williams has told him. He says she told him that the kids were surprised while "sportin'" and took ill. He says that, on the first day that Hale came to town, Abigail told him (Proctor) that the girls' illness had nothing to do with witchcraft.