When air is piped in from outside and into a pressurized cabin, it is also dehumidified in the process. All airlines provide beverage service on even short-haul flights to combat the dryness of the air. Why is it necessary to dehumidify cabin air?
A lot of consideration is placed on the airframe (which has to hold for 20+ years) rather than the passengers.
One of the primary reasons is to slow corrosion of the aluminium airframe. This is also why the Boeing 787 is a lot more humid - corrosion is no longer an issue in a composite fuselage (or a lot less at least).
the greatest issue lies with the cold fuselage skin interacting with moist air, which will condensate and speed up the rate of corrosion.
But condensation in the gap between cabin and hull can be lethal. This gap contains much of an aircraft's wiring, and water can damage that wiring's insulation. Such a problem is thought to have contributed to the loss of a Swissair DC-11 off the Atlantic coast of America a few years ago. Too much condensation causes other difficulties, as well. There have been cases when ice has built up inside the tailplanes of aircraft, causing their rudders to freeze and thus preventing their pilots from steering them.
An aircraft such as a Boeing 747-400 can accumulate as much as 700kg of condensation before it reaches equilibrium. Even though that is only about 0.17% of the 400 tonnes such an aircraft weighs fully laden, it is enough to cause problems for the pilot when he tries to trim the balance of the aircraft. And 700kg is about the weight of nine men. So the aircraft is carrying the equivalent of at least nine non-fare-paying passengers, in an industry where every extra kilogram affects fuel consumption and profit margins.
There are a couple of things that contribute to the dryness of cabin air:
Cabin air is drawn from engine bleed taps in the high pressure compressor. This is ambient air, which at the flight levels is normally quite dry and then heated and compressed, which will lower the relative humidity, making it seem even drier.
The air cycle machine subsequently extracts moisture from this hot, high pressure air within the pack and used it to provide evaoprative cooling in part of the pack responsible for cooling the air flowing through it (The pack needs to extract heat so when air is expanded into the cabin it is cool enough - it does this with compression/expansion cycles and cooling).
As the air coming out of the packs undergoes expansion to cabin pressure, it will cool down and this will help counteract some of the dry feeling by boosting relative humidity.
The primary reason low humidity is desirable in most airplanes is that water contributes to corrosion and that is bad for the airframe.
The Boeing 787 takes a new approach and provides pressurization and air conditioning with electric compressor motors instead of engine bleed air, and using this method (and a composite airframe) allows for much higher relative humidity in the cabin.
When I flew the C-141 and C-17, their environmental systems weren't intentionally dehumidifiers (designed to reduce the overall humidity), but they did push the air through a fabric bag called the 'sock' to capture droplets that formed during cooling. As air continued to flow, that moisture could evaporate and rejoin the stream, or be vented overboard through a drain.
At altitude, the air is very dry. Flying through weather in the approach/departure environment could cause a lot of moisture to get ingested, and so needed to be sorted out.
To add to the answers above: the other main reason (besides condensation) for keeping humidity low is to prevent the growth of mold and bacteria in the cabin and the air conditioning system, which over time may pose a health hazard to the plane's occupants.
And I can't stress enough that the air is not de-humidified, but that the outside air (which is pumped into the fuselage to provide pressure and fresh air) at cruising altitudes is bone-dry. The air in the cabin is actually more humid than it would otherwise be, thanks to the passengers (when they exhale or sweat they release water vapor into the air, and since about 50% of the cabin air is re-used it helps to keep the humidity levels a little higher than they would normally be).
The newest-generation of widebody planes from Boeing and Airbus (787 and A350) actively humidify the air in the cabin, but that requires carrying extra water on every flight (50-100kg, according to Flight Global, plus the weight of the humidification equipment). Additionally, these airplanes need to install active drying systems in the the areas outside the cabin (e.g. behind the lining or above the cabin ceiling) to prevent the insulation mats from soaking up all that extra humidity (example from CTT). In conclusion, the air will be more humid on these aircraft, but it takes a lot of effort (and weight).