This term is used in the context of the Cassiopaea material and QFS studies to denote an expanded view of reality. This is the larger reality in which the reality of three dimensions of space and linear time is embedded. The hyperdimensional reality is not directly observable from inside itself. We can however see shadows of hyperdimensional contents and principles through their effects, a bit like the spectators in Plato's cave analogy.
A comprehensive description is not possible but drawing on all the material brought together by the QFS from esoteric tradition, direct experience and different sciences, we can list probable attributes of this broader reality:
Multiple Levels - Reality seems to be constructed of worlds within worlds, each offering a more all encompassing view of phenomena and beings. This idea corresponds to the 4th Way ray of creation and the Cassiopaea/Ra scale of densities. The human level of perception is but a slice of a more complex world. We can speak of archetypal forms having a concrete existence in a hyperdimensional context from which they are projected onto the level of human experience.
Cyclic Time - Time is probably perceived as non-linear and consisting of cycles within cycles from the hyperdimensional perspective. Causality is not a strict one-way relationship between pasts and futures but rather an interactive probabilistic joining of possible pasts and futures into an expanded present. This does not mean that time would not exist but that it is not limited to one dimension.
Consciousness and Energy as Commodities - Consciousness appears to be a construction material as well as a sort of food or commodity for certain beings. From the perspective of the hyperdimensional reality, emotions, thought and experience have a certain solid substance, as does knowledge. These are items to be either hoarded or passed around, as in STS vs. STO.
Variable Physicality and non-Physical Existence - Legends and the UFO phenomenon for example feature a large assortment of variously ethereal and physical beings. Consciousness is not limited to manifesting through a solid physical body.
The term hyperdimensional reality is very general. For more details on how it might be constructed, see Densities and Worlds.