A brace and bit is a hand tool used to drill holes. Grandpa knows a trick that is effective in catching raccoons, and all he needs to do it is a brace and a bit.
When Billy gets his pups, he realizes he needs a coon skin to teach them to hunt, but, try as he might, he cannot catch a coon. Grandpa tells him about an old trick he used as a child that rarely failed in helping him get one of the ring-tailed animals. Grandpa gets a brace and a bit about an inch and a half in diameter. He tells Billy to go down by the river where the coons are known to live, and drill a hole about six inches deep in a log. At the bottom of the log, Billy needs to put something shiny, like a piece of tin, and then, around the hole made by the brace and bit, he should pound nails in about an inch apart and at a slant, so that the points stick out about halfway down the hole. The raccoon will then be drawn by the shiny tin, and, curious, stick his hand in to grab it. Because his hand will be closed around the piece of tin, it will be larger than when it was thrust into the hole, and because of the nails, the raccoon will be unable to get his hand out. It will not occur to the raccoon to let go of the tin so that he can extricate his hand; he will be trapped, and Billy will have his raccoon.
Billy follows his grandpa's instructions carefully, and makes fourteen traps. For the first week, he catches nothing, and is angry and discouraged, thinking his grandpa had been fooling him. Billy's father, however, points out that perhaps Billy has left too much of his scent around the trap area. Sure enough, after about a week, the scent has worn off, and Billy catches a large coon, "trapped by his own curiosity" (Chapter 7).