How did Bud feel when Miss Thomas tells him that he will be staying with them?

Miss Thomas tells Bud that she wants him to stay in chapter 16. Bud's immediate reaction is a "gigantic smile" which he says "split [his] face in half." This implies that he is very happy. The metaphor, in which the smile is described as splitting his face "in half," emphasizes how broadly he is smiling and thus how extremely happy he is.
He also reassures Mis Thomas that he will be strong enough, in body and in spirit, to help and endure Mr. Calloway. Bud is then given a new suitcase, of which he is proud, and he joins in the laughter when Miss Thomas makes a joke about the car. This suggests that he is pleased and also that he feels comfortable with his new family.
Bud's delighted reaction to being told that he can stay with his new family is understandable. He has, up to this point, led a rather nomadic, chaotic, and unfortunate life. Indeed, the story begins with Bud as a ten-year-old boy, about to be placed with his third foster family. His mother has died and he has never known his father. Miss Thomas offers him the opportunity for a more permanent, secure, and happy life.