Harry's love for the arts is well documented - from his official photos at Eton, to reports of his own personal collection.
The 'in crowd' always seem to have a favourite hangout, and I think the Arts Club is the 'in' place at the moment. I can see the appeal for Harry (and other Celebs), as it is very private, and hosts a variety of different kinds of events. It seems to be more a mature evening out than that offered at, say, Mahiki.