To build on a steep cliff there is no level ground for a foundation, so holes are cut into the rock, one end of a crossbeam is half-inserted and the other end projects outward to carry the hall beyond it, and wooden posts are then set against the cliff to brace it.
The Hanging Temple at Mount Heng was built in this way. It was begun around 491 (Taihe 15 of the Northern Wei), its halls cantilevered over the cliff on beams half-sunk into the rock.
The halls of the Hanging Temple seen today were largely rebuilt in the Ming and Qing, and they continued the tradition of cantilevered timber cliff architecture.
So "hanging" in this class means that the building hangs outside the cliff, held by beams from within it, and not that it has nothing to rest on.
Cliffside Architecture is the class in the chapter on Architecture on buildings set against cliff faces. It takes the Hanging Temple at Mount Heng as its example: begun around 491 (Taihe 15 of the Northern Wei), with halls carried on beams half-sunk into the rock and braced with posts standing against the cliff; the halls that survive were mostly rebuilt in the Ming and Qing, continuing the tradition of cantilevered timber building.