Hester was trying to plead her case, so that she could keep Pearl. Minister Dimmesdale spoke up in her defense and persuaded everyone else in the room to allow Hester to keep Pearl because God gave Pearl to Hester.
Chillingworth has become Dimmesdale's physician. They have been spending a good amount of time together because Chillingworth is trying to figure out what Dimmesdale is hiding.
Chillingworth finds Dimmesdale asleep. While Dimmesdale was resting Chillingworth pulls back his shirt to find out the reason why he constantly holds his chest and is plesantly surprised by his discovery.