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Two reasons:
1. Reset routine
Fed has very few obvious ones - his on-court personality is based on being effortless, but shaking off his hair and blowing at his fingers are the two more consistent ones.
In tennis a "reset routine" is what a pro (or college) player does between every point to (a) forget about the previous point (b) get their focus ready for the next point.
For Lleyton Hewitt its tugging away at this strings (even though he uses Luxilon poly strings which don't move at all).
We know Nadal's legendary reset routine(s) ;-) Players also have routines for the changeovers between games. I think this is the primary reason why Fed does it.
2. Keep away blisters - this is mostly an old school thing.
Mardy Fish did this as well. With cushier modern grips blisters on fingers are rare when using the correct-sized grip (which Fed does) and more commonly caused by using a smaller-than-normal grip size (to aid in exaggerated topspin).
See Rafael Nadal or Gael Monfil's perenially taped up fingers for reference.
Fed, rarely, if at all, has had any visible sign of blisters Not related but the lack of his pre-match taping routine is also legendary, all part of the "effortless Fed" persona.
While many pros on the tour tape their ankles and wrists (Nadal and many others) or wear ankle braces (Murray), Fed is renowned for no tape and a very simple routine - he does some stretches in the locker room and is ready to go.
This is part of the myth/image - in reality all the warming up/hard work he does, is probably behind the scenes.
Sweat, as others have suggested *may* be a factor but he always has a wristband on his right hand and fingers don't really sweat that much (unless one is Andy Roddick or John Isner, both of whom sweat buckets).
Fed, if anything, comparatively sweats much less than other pros (did I mention the "effortless Fed" persona ;-) and the locker room joke is that after his matches, Fed looks like he's walked out of a salon (no sweat, dry clothes, hair in place)