Why won't Walter Cunningham take the money from Miss Caroline?

Everyone in Maycomb knows the Cunninghams are a very proud family. They may be dirt poor, but they are disinclined to accept any free handouts. They will never accept any favors from anybody unless they can repay it. Since they can rarely afford to pay money, the family repays people in the products of their labor.
Since Walter knows that this philosophy is part of being a Cunningham and realizes that he will not be able to repay the quarter on offer, he refuses to take it. Miss Caroline is new to the town and does not know of their tradition. She sees Walter's response as rude and is quite annoyed with him. Scout is urged to enlighten Miss Caroline about their ways and when she does so in Walter's defense, Miss Caroline, who had already been irked by her on a previous occasion, decides to punish her.
It is for this reason that Scout later rubs Walter's nose in the sand; she believes that he brought trouble upon her. Jem saves the boy from his sister's vengeance and invites him home for dinner. In this instance, Jem displays the qualities Atticus has inculcated in his children. Scout is still too young and naive to understand her father's philosophy and responds on an emotional level.
It is clear that Atticus has taught his children not to discriminate, and Scout's action in this particular situation in some way foreshadows her intervention later in chapter 15, when she speaks to Walter's dad in front of the jailhouse where Tom Robinson is incarcerated. Mr. Cunningham is part of a lynch mob intent on removing Tom and executing their brand of justice. Her father has been aware of the threat and is outside the prison, keeping watch, when they arrive.
Scout recognizes Mr. Cunningham and goes to speak to him about his son and his entailment. Her action prevents a serious confrontation and probably saves both her father and Tom from serious harm. Mr. Cunningham responds to her and asks the mob to leave, which is exactly what they do.

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.