Because of the way he has been brought up, Walter Cunningham won't take any lunch money from Miss Caroline.
As Scout explains in the book, Walter is a Cunningham, and in Maycomb, the Cunninghams are known to be a poor but proud family. A Cunningham will refuse a gift he cannot readily repay.
"...The Cunninghams never took anything they can’t pay back—no church baskets and no scrip stamps. They never took anything off of anybody, they get along on what they have. They don’t have much, but they get along on it."
So, despite not having any lunch, Walter refuses to take the lunch money offered by Miss Caroline. Scout explains to Miss Caroline that Walter will never have the money to repay her, and since she can't use any "stovewood," forcing Walter to take the money would be wrong. In regards to the "stovewood," Scout is referring to the way the Cunninghams usually pay for any services or products.
Scout recalls a time when Walter's father had some legal work done by Atticus. Accordingly, Mr. Cunningham later paid Atticus with stovewood, hickory nuts, turnip greens, and Christmas holly. Scout tries to explain to Miss Caroline that the Cunninghams have their own way of doing business, but Miss Caroline refuses to listen. In the end, Scout is spanked for her troubles, a punishment she considers unjust.