Why does only class C airspace have an "outer area"? There is an area called "outer area" near class C airspace. As described by this AIM:


  Though not requiring regulatory action, Class C airspace areas have a procedural Outer Area. Normally this area is 20 NM from the primary Class C airspace airport. Its vertical limit extends from the lower limits of radio/radar coverage up to the ceiling of the approach control's delegated airspace, excluding the Class C airspace itself, and other airspace as appropriate. (This outer area is not charted.)


As it is not charted and there is no FAA regulations about outer area, I would think it could also be set near class B or D airspaces. Why don't any other airspaces but only class C airspace has the outer area?
The Class C outer area is the airspace surrounding Class C airspace out to 20nm, and from the lower limit of radar coverage to the top of the controller's airspace.  The outer area is not charted, although you can see in the white boxes (in the chart below) adjacent to Class C airports that you can initiate communication within 20nm.

ATC offers Class C services (e.g., flight following) on a workload-permitting basis to participating aircraft operating in the outer area, and aircraft in 2-way radio communication will receive IFR/VFR aircraft separation services.
So you can think of Class C airspace as 3 concentric rings: the 20nm outer area, the 10nm shelf area, and the 5nm surface area.
Source AIM 3-2-4