Why did the author most likely use the phrase "the thermometer stood at sixty" in the first paragraph of the text?

This phrase is most likely used to help establish the setting of the scene. The author is indicating that the weather is pleasant—it is holding steady at sixty degrees (Fahrenheit, of course), which is a very comfortable temperature, especially early in the day. This is substantiated by Mr. Nilson’s own thoughts when he sees the Japanese Quince in blossom in the Square Gardens: “’Perfect morning…spring at last!’” These sorts of platitudinous issuances about the weather pepper the story throughout, and serve as a cover in conversation, sharply contrasting with the true discomfort felt by the two characters upon meeting each other in the Gardens. Both had been under the impression that he was the only man out on such a beautiful, unseasonably warm morning, and the shattering of that impression left each of them feeling as if he had been intruded upon. So, while the weather is exceptionally nice outside, within both of the men is an uneasy feeling, as though each has had his sense of individuality stripped away.