explain specifically why had billy lost his confidence for this very short time

It's difficult to know exactly where you mean but in Chapter 8 Billy cuts down a big tree where a coon is trapped. His dog keeps him motivated.

when Billy had bet with Rubin.During the the hunting he almost lost his confidence but his dogs motivated him. I want to know how the dogs motivated Billy.It's in chapter 13.

Billy and the Pritchards approach the isolated short, broad tree in which the ghost coon is treed. Billy climbs the tree, but cannot find the ghost coon and comes back down. The dogs keep searching, but after Billy climbs the tree again and still cannot find the coon, they all give up. Billy pays Rubin the money for the bet.

Just before they leave, Little Ann catches the scent in the breeze. She bawls by a nearby large gatepost, and Old Dan joins in. Billy gets on Rubin's shoulder and sees the post is hollow. He sticks a switch into the post and hits the coon. It jumps out, and the dogs fight him. The coon is elusive and scrambles up the original tree. Billy climbs up, and the coons cries - it knows its end is coming.