"Hanging" is a butcher's term. Meat in a butcher shop is hung in a walk-in box (a large, walk-in refrigerator) and hung from metal hooks to age. Most people around here call it "letting the chicken rest." You simply put it in the fridge for the amount of time you want. The range that I have read is typically around 1-3 days. This is to allow the rigor mortis to pass, which in turn will make the meat more tender. I have no experience yet with poultry yet (a few more weeks!) but with beef, lamb, venison, etc. it makes for better tasting meat as well.
I just started doing this. I get back from processing and put the birds in the fridge for 24 hours min (sometimes 48), then freeze. I have found that the meat is more tender. They are in their little vac-packs all processed, just sitting in the fridge.
Yes, the chickens are processed before aging, i.e. feathers and innards removed.
How it is aged is not uniform. Some people use a brine, some seal and put in the fridge, still others believe the chicken should be exposed to the air while aged (and have specific requirements for the humidity of said air).
To me, the strange part in all of this is that it used to be common in the US (and still is in many parts of the world) to kill a chicken and eat it the same day.