Theory:

“I know how it will be,” his sister had said when he was preparing to migrate to this rural retreat; “you will bury yourself down there and not speak to a living soul, and your nerves will be worse than ever from moping. I shall just give you letters of introduction to all the people I know there. Some of them, as far as I can remember, were quite nice.”
 
Framton wondered whether Mrs Sappleton, the lady to whom he was presenting one of the letters of introduction, came into the nice division.
Explanation:
 
The third paragraph of the story introduces one of the minor characters: Mr Nuttel's sister. Moreover, the paragraph also provides more insight into Mr Framton Nuttel's situation.
 
Through a flashback, the narrator provides a crucial backstory to Mr Nuttel's visit to the "rural retreat". Flashbacks are at times used in literature and other art forms to bring depth and understanding to a story. Here, Mr Nuttel's flashback helps us understand how and why Mr Nuttel came to Mrs Sappleton's residence. The flashback technique is otherwise known as analepsis.
 
In Mr Nuttel’s flashback, we could hear what his sister had to say when he was prepared to migrate to this countryside. She was worried that he would choose not to socialise. Rather than interacting with people, he 'would bury himself down', with his nerves turning worse.
 
The phrase "you will bury yourself down there" indicates Mr Nuttel's tendency to get into his shell, i.e., become withdrawn and invisible. As observed previously, Mr Nuttlel was more comfortable keeping to himself; however, he also suffers from psychological problems such as anxiety and depression. Hence, it is quite important that he talks to people so as to keep a steady and healthy mind. The word "moping" also suggests that depression was indeed one of Mr Nuttel's ailments. He was probably moving to the countryside to bring his mind some fresh air and enjoy a peaceful life.
 
Coming back to the flashback, we also see how the sister had suggested a solution for his situation. She had said that she would write him some letters of introduction to everyone she knows there. As it turns out, the country to which Mr Nuttel was migrating was a place familiar to the sister. On the other hand, the line also reveals that the place was new to Mr Nuttel. Hence, his sister believed that her letters would help him get acquainted with the residents and might help him find friends. She also added that some of the people she knows there were friendly and helpful.
 
With this ends Mr Nuttel's flashback, and the story is brought back to Mrs Sappleton's residence.
 
As Mr Nuttel was thinking about what his sister had said about the people, he wondered whether Mrs Sappleton, the lady to whom he was handing one of the letters of introduction, belonged to the nicer division. 
 
Meanings of difficult words:
 
S.No
Words
Meanings
1
MigrateMove from one place to another
2
RuralIn, relating to, or characteristic of the countryside rather than the town
3
Retreat
A quiet or secluded place in which one can rest and relax
4
Moping
To become unhappy and dejected; to be unhappy and unwilling to think or act in a positive way, especially because of a disappointment
Reference:
National Council of Educational Research and Training (2008). The Open Window – Saki (pp. 55 - 61). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.