Walter Cunningham follows Jem and Scout home for lunch because Jem has invited him to join them for the noon meal.

Walter Cunningham comes from a poor but proud family.  They are hard workers, and will be beholden to no one, paying their debts faithfully in whatever way they can.  Walter comes to school each day with no lunch, because he cannot afford one, and in his family, it is better to go hungry than to lose one's dignity.  Miss Caroline, his new teacher, notes that he has no lunch and offers to let him borrow a quarter to buy some.  When Walter politely refuses to take the quarter, Miss Caroline is nonplussed, until Scout volunteers the explanation that "the Cunninghams never took anything they can't pay back...they don't have much, but they get along on it".  As Scout has already gotten off on the wrong foot this morning, Miss Caroline insensitively takes her remarks as impertinence, and disciplines her (Chapter 2).

Later, at lunch, Scout is taking out her frustrations on poor Walter for, as she sees it, getting her in trouble.  When she explains what happened to Jem, he immediately understands the situation, and invites Walter to join them for lunch.  Walter at first does not know how to respond, and since time is running short, Jem and Scout start out without him.  After thinking about it for awhile, Walter decides that since Jem's offer is a neighborly invitation and not charity, it is all right to accept, and he runs after Jem and Scout, calling "Hey, I'm comin'!" (Chapter 3).

In Harper Lee's To Kill a Mockingbird, why does Jem invite Walter Cummingham for lunch?

In To Kill a Mockingbird by Harper Lee, a new teacher has joined the school faculty.  Miss Caroline Fisher will be Scout’s first grade teacher.  Miss Caroline makes many mistakes her first day because she is new to teaching. 

Miss Caroline offers to lend Walter Cunningham money for his lunch.  She believes that he has just forgotten his money.  Scout has always been free to talk and say what comes to her mind.  In school, the rules change.  Scout knows that the Cunninghams have no money, and they do not like to owe money to other people.  When Scout tries to explain about Walter to Miss Caroline, the teacher punishes Scout.

Scout finds Walter on the playground and begins to beat him up.  Jem sees her and stops the fight.  He asks Walter to come home with them for lunch.  Scout does not understand why she should not be mad at Walter for causing her to be in trouble with the teacher.

It is obvious that Jem is moving toward maturity.  He understands the problems of the Cunninghams.  Atticus has done legal work for them.  Mr. Cunningham has always paid him, not with money but with goods from his land.  Jem also realizes that Walter cannot help from where he comes or what his parents do. He treats Walter with respect. 

Calpurnia is glad to have Walter eat lunch with them.  Walter covers his food with maple syrup at which point Scout makes a face and rudely comments on his eating habits.  She is sent from the table and scolded by Calpurnia.

“There is some folks who don’t eat like us, she whispered fiercely, but you ain’t called on to contradict ‘em at the table when they don’t. That boy’s yo’ company and if he wants to eat up the table cloth you let him, you hear?”

“He ain’t company, Cal, he’s just a Cunningham.”

“Hush your mouth! Don’t matter who they are, anybody sets foot in this house’s yo’comp’ny, and don’ you let me catch you remarking’ on their ways like you was so high and mighty!”

To further her understanding, Atticus teaches Scout one of the primary lessons of the story. A person should never criticize another person until he walks around in their shoes or looks at things from their perspective. This lesson will help Scout later in the novel when she tries to understand the unfair and ill treatment of others.