The conflict in a story is the element in the narrative which presents a struggle or challenge to the protagonist. This is a very gentle story, but it certainly has a conflict. Miss Esther Porley, the protagonist, has resolved at the beginning of the story to open up her house to a fellow lodger. It has been too long since her mother died and she feels that she is not being sufficiently generous with what she has: she wants to open up her house for the use of someone who lives in the city and might have need of country air. She specifically states that she would not like a "girl" or a rowdy young man, and expresses a preference for an older person—meaning an old lady. This message is passed along to the Committee in town, but unfortunately, this committee does not know Miss Esther, and they do not interpret her wishes as she had expected.
The conflict in the story, then, comes when Miss Esther Porley arrives at the station, expecting an old lady, and finds nobody there but "one clean-faced, bent old man with a bird-cage in one hand." This immediately fills Esther with apprehension. She had not expected a man, and indeed she is anxious about what others on the platform will think of her if she approaches this old man unaccompanied. The story is set in a time period where men and women did not generally mix without chaperones unless they were married, so the presence of Mr. Rill instead of the expected old lady presents Miss Esther with a considerable challenge; indeed, she is "enraged with the Country Week Committee."
Ultimately, however, although Esther has not had a man in her house for "fifteen years, at least," she decides to give Mr. Rill a chance and invite him into her house. The result of this decision on her part proves to be momentous, as she finds him to be not at all like "other men," and when the time comes for him to leave, neither party is at all happy about it. At the end of the story, Miss Esther confides that Mr. Rill has determined to come back to live with her permanently–and has left his bird at the house as if as a promise.