Why does Bud keep the extra trousers from the Sleet family in Bud, Not Buddy?

There is a fairly simple answer to your question found in chapter 11 in Bud, Not Buddy.   Bud keeps the extra trousers that the Sleet family gives him because he is sick of wearing short pants that he calls “knickers.”  “Trousers” and “knickers” are typical words in the 1930s to describe long pants and short pants. 

This scene happens when Mr. Lefty Lewis takes Bud to his daughter’s house.  After Lefty Lewis introduces his grown daughter as Mrs. Sleet, Bud is given a pair of pants used by Mrs. Sleet’s son.  Because Mrs. Sleet’s son has grown too old for the trousers, they are the perfect set of pants for Bud.  Bud is ten years old now and considers knickers to be for a small child.  As a result, he is tired of wearing them all the time.  Therefore, Bud is happy to receive the hand-me-down pants.  Bud is invited downstairs to breakfast after he exclaims, “No more knickers!”