The hunt you are referring to is the runoff of the hunting contest. Eliminations are over, and Billy's dogs have made it to the finals, along with two other sets of dogs. The two other teams who have made it to this point are "four of the finest hounds there are." One of the teams, a pair of handsome walker hounds, have already won four gold cups in other contests.
When it is Billy's time to turn his dogs loose, he unties their ropes and holds their collars for a moment, pulling them close and whispering to them, "This is the last night. I know you'll do your best." When he lets them go, the dogs run towards the woods, but stop just as they reach the dark shadows, turn around, and stare straight at Billy for an instant. The judge, noticing this, recognizes that an extraordinary communication has occurred, and asks Billy what Old Dan and Little Ann were saying to him. Billy responds,
"Nothing that anyone could understand, but I can feel that they know this hunt is important. They know it just as well as you or I" (Chapter 16).
Billy is so close to his dogs that he seems to be able to communicate with them in a way that is almost uncanny. When he tells them that the hunt has great significance, they somehow sense what he is saying, and stop for a moment to let him know they understand.