The direct context of the quote is that the retired Woodifield, who has had a stroke, is only allowed out to visit the City--central London--once a week by his wife and daughters. They can't imagine what he does there, but then the narrator explains that "we cling to our last pleasures as the tree clings to its last leaves."
A tree clinging to its last leaves is an image of a tree near to death. This statement implies that Woodifield too is nearing death. He is an older man, and it does give him pleasure to get out and visit an old friend, as well as smoke a cigar.
By more than simply a reference to Woodifield, it sets the tone for a very bleak story which primarily concerns death: the death of Woodifield and the boss's sons in World War I (Woodifield has come to report...
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