Theory:

“Do you know many of the people round here?” asked the niece, when she judged that they had had sufficient silent communion.

“Hardly a soul,” said Framton. “My sister was staying here, some four years ago, and she gave me letters of introduction to some of the people here.”

He made the last statement in a tone of distinct regret.
Explanation:

As Mr Nuttel was lost in his thoughts, Vera decided to intervene and reinitiate the conversation. She began by inquiring whether he knows many of the residents from the area.
 
The line "asked the niece, when she judged that they had had sufficient silent communion" employs humour- and irony in particular. The word "communion" refers to exchanging thoughts, yet Mr Nuttel was rather engrossed in his own. Though Mr Nuttel was silent, he was nowhere close to "communicating", as the line had declared. Hence, the phrase "they had had sufficient silent communion" becomes ironic as the real meaning is opposite to what was presented.
 
Moreover, the line also suggests that, unlike Mr Nuttel, Vera is not comfortable being quiet and is impatient to break the silence.
 
Back to the lesson, Mr Nuttel responds to Vera's question by observing that he doesn't know anyone from the place. He also tells her that his sister was staying there around four years ago. Moreover, he adds that his sister had offered him the letters of introduction to several of the folks there.

He was visibly unhappy with the whole act of 'meeting the strangers', for the narrator describes how Mr Nuttel's final words about "the letters of introduction" was filled with a strong sense of regret.
 
Speaking of the above conversation, it is worth observing how the minor details revealed to Vera (such as Mr Nuttel's unfamiliarity with the place and the time of his sister's stay) would play a crucial role in the rest of the story. 
Meanings of difficult words:
 
S.No
Words
Meanings
1
CommunionThe sharing or exchanging of intimate thoughts and feelings, especially on a mental or spiritual level; a kind of communication
2
HardlyUsed to qualify a statement by saying that it is true to an insignificant degree
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.