Scout rubs Walter Cunningham's nose in the dirt because she thinks that it is his fault that she got in trouble with Miss Caroline, her first grade teacher.

After having already gotten off to a rocky start when Miss Caroline reprimanded her for being too advanced in reading (a skill which Miss Caroline attributed to the "inappropriate" and unskilled intervention of Atticus), Scout's situation only worsens when she attempts to rescue Walter from certain humiliation. Miss Caroline notices that Walter did not bring a lunch to school and insists that she will give him a quarter to purchase one as long as he pays her back the next day. Hearing this, Scout intervenes and explains that Walter is a "Cunningham," which means that he won't accept the quarter from Miss Caroline because he will not be able to pay it back; he is too poor to do so, just as he is too poor to bring a lunch.

Unfortunately for Scout, Miss Caroline is sick of her commentary; she punishes Scout my making her stand in the corner—an act of humiliation for the young girl. Thus, Scout gets even by catching Walter in the schoolyard and rubbing his nose in the dirt.

The answer to this question can be found at the very beginning of chapter 3:

Catching Walter Cunningham in the schoolyard gave me some pleasure, but when I was rubbing his nose in the dirt Jem came by and told me to stop. “You’re bigger’n he is,” he said.

“He’s as old as you, nearly,” I said. “He made me start off on the wrong foot.”

“Let him go, Scout. Why?”

“He didn’t have any lunch,” I said, and explained my involvement in Walter’s dietary affairs.

The section about Scout explaining Walter's dietary affairs can be found in chapter 2. Miss Caroline tried to help Walter out with his lunch by loaning him a quarter, but Walter wouldn't take it. Scout tried to explain that Walter wouldn't do that because he is a Cunningham. Miss Caroline doesn't understand, so Scout attempts to explain better; however, her explanation comes across as slightly insulting to the teacher:

You’re shamin‘ him, Miss Caroline. Walter hasn’t got a quarter at home to bring you, and you can’t use any stovewood.

Miss Caroline then punishes Scout, and Scout is completely humiliated in front of her peers:

Miss Caroline picked up her ruler, gave me half a dozen quick little pats, then told me to stand in the corner. A storm of laughter broke loose when it finally occurred to the class that Miss Caroline had whipped me.

Chapter 3 begins with Scout rubbing Walter's nose is the dirt because it's her form of revenge. Scout feels that Walter is ultimately at fault for her humiliation.

She does this because she is mad at him.  She is mad at him because she feels like it is his fault that she got in trouble with her teacher, Miss Caroline, on the very first day of school.

Walter is poor and so he was not able to bring lunch.  Scout tried to explain that to Miss Caroline when she kept asking Walter about it.  That (along with Scout being able to read) made Miss Caroline angry and she hit Scout's hands with her ruler.  It didn't hurt, but it humiliated Scout and so she wanted revenge.

The incident occurs at the beginning of chapter three. Scout has had a bad first day at school, starting with Miss Caroline criticizing her father for helping her read. Scout was offended by this criticism and was quite upset about it. Jem's attempt at making her happy during their break didn't succeed much. The second incident happened when she started writing a note and Miss Caroline told her to stop, repeating her original injunction that Atticus should not teach her. 

Miss Caroline is clearly annoyed with Scout, and when Walter Cunningham refused her generosity when she offered him money to buy lunch since he did not have any, Scout attempted to explain the Cunningham tradition of not accepting anything if they could not pay it back. Miss Cunningham had, by this time, obviously had enough of Scout and decided to punish her. She gave her six pats on the hand with her ruler and sent her to stand in the corner.

Scout, when second break came, was obviously upset that her defense of Walter Cunningham had culminated in the punishment she had been administered. This was, evidently, the cherry on top and she had to vent her frustration. She therefore held the innocent boy responsible for the suffering she had experienced and wanted vengeance. It is for this reason that she decided to rub "his nose in the dirt." When Jem intervened and stopped her bullying, she told him that Walter had made her "start off on the wrong foot."

Jem later magnanimously invites Walter over to their house for dinner, probably to make up for Scout's unfair abuse.

This is because he made her "start off on the wrong foot with her first grade teacher, Miss Caroline.

Scout got in trouble with Miss Caroline partly because she was trying to explain to the teacher why Walter did not have any lunch.  Scout was trying to explain to the teacher that Walter was too poor to have lunch.  Miss Caroline got mad at Scout because of this.

Miss Caroline is mad because (in my opinion) Scout makes her feel like she does not know what she is doing on her first day at this new school.

Jean Louise Finch, “Scout”, blames her mishaps that morning on Walter Cunningham. There was an incident when Miss Caroline, the school teacher, offered a quarter to Walter for lunch. Scout tried to explain to Miss Caroline that the Cunningham’s didn’t borrow money because they had no money to pay the debt back. Her explanation was lost on the teacher, and after several other issues Scout was involved in, caused her to get her hand smacked as punishment. These events, eventually lead to her transference towards Walter.

Scout rubbed Walter Cunningham's nose in the dirt because she blames him for getting in trouble on her first day at school.

When Scout's teacher, Miss Caroline, offers to give Walter lunch money because he doesn't have any, Scout explains that he won't be able to pay her back, and that he won't take the money. The Cunninghams are a proud family who refuse to take charity.

Miss Caroline becomes upset and frustrated with Scout's explanation and slaps her hand with a ruler. As a result, Scout rubs Walter's nose in the dirt for starting her off on the wrong foot with the teacher.

Because he was involved in her getting in trouble that day in school. Because he wouldn't explain himself, Scout did, only to get shamed in the process.