Grandpa puts his britches on backwards because he is excited and gets dressed too fast.
Grandpa encourages Billy to enter Old Dan and Little Anne in a coon hunting contest. He goes with Billy, and Billy’s father and the judge accompany them. The dogs scent a raccoon.
There was a mad scramble. Grandpa tried to put his britches on backwards. The judge and Papa ran over to help him with his shoes. Each one tried to put a shoe on the wrong foot. I was laughing so hard I could do nothing. (Ch 16, p. 186)
The dogs had found the coon in swampland. Old Dan came to a tree, but there was no coon. The judge decided the dogs had been fooled. They find the coon, and eventually Billy wins the competition.
The contest demonstrates that Billy has a strong bond with his dogs, but also his father and grandfather. All of them support him, and they all love each other.