Applications
Keep in mind that this is merely a very general list of the abilities that a higher-dimensional entity would possess, and is not necessarily applicable to all of fiction.
Pseudo-Invulnerability: Due to the additional axis spatially inherent to them, higher-dimensional objects are comprised of more information than what can normally be described in lower-dimensional spaces, and thus may only be capable of interacting with them through lower-dimensional "slices" or cross-sections of their bodies.
Pseudo-Teleportation: Higher-dimensional entities may be capable of seemingly phasing in and out of sight by moving through additional axes imperceptible to lower-dimensional beings, granting them a notable advantage in overall range and mobility.
Unconventional Physiology: Although much of the same physical laws and principles would theoretically apply to higher-dimensional worlds, they would likely be instantiated very differently, resulting in matter behaving in strange ways compared to our own Universe.
Large Size: Due to the additional angles possessed by them, higher-dimensional objects and beings would be much larger than lower ones.