A palm room is an exotic and elegant lounge or dining room, decorated with lavish palms on the inside.
An interesting history of palm rooms can be found in this excerpt from Trees in Paradise: A California History by Jared Farmer:
In the Anglo-American world, the pad for potted fronds – the Mediterranean in miniature – reached its zenith in the Edwardian era. Every luxury hotel worth its name added a "palm court" or a "palm room," an ornate lounge or banquet hall or tearoom with upholstered furniture, a glass ceiling, painted murals in "Pompeiian" style, and tropical-looking plants. To the strains of a "palm-court orchestra" – violin, cello, and piano playing light favorites from Dvořák, Grieg, Herbert, and Lehár – patrons sat for high tea under the fronds. The trend began in establishments like the Carlton, the Waldorf, and the Cecil in London and the Plaza, the St. Regis, and the Commodore in New York, and spread to the leading hotels of California and many states between. The "great white fleet" of the United Fruit Company added palm rooms to its Panama liners. The premier cruise ships of the era, the trio of Olympic-class vessels from the White Star Line – RMS Olympic, RMS Britannic, and RMS Titanic – contained lavish, oversized palm courts that combined verdure with opulence.
- the Edwardian era is the first decade of the 1900s,
- according to Farmer, this trend seemed to reach its peak in that same time span, and
- this is also the time frame when O. Henry accomplished a lot of his writing
it appears that the character Robbins had developed quite a penchant for spending time in the posh palm rooms of luxurious resorts. Based solely on your one-sentence excerpt, I imagine the rotund fellow was probably elegantly dressed on such occasions.
Palm court on the Titanic