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.