Before I get into what higher dimensions can do in post apocalyptic settings (among others), I first want to explain the difference between a dimension and a plane. A plane is an alternate universe, usually an afterlife in d20 terms. Extradimensional spaces, such as portable holes, are actually minature planes. Dimensions are mathmatic directions that include, but are not limited to X, Y and Z (length, width and depth). Time has been called the fourth dimension, but that is something else (and as I am not educated in that type of math, I have no idea how to explain it). If you want to read how to incorporate creatures from higher dimensions into your d20 games, Kobold Quarterly 8 has a very nifty article on the subject.
So, higher dimensions are those places that exist in the setting's universe but can not be accessed by three dimensional creatures. In face each dimension above the second may be completely seperate from the others or may have influences that are difficult to explain (does dark matter exist in the 5th dimension?). But the ancients in Gamma World and Mutant Future may have been able to access them through technology. Of course human minds can not comprehend what life might be there, much less existence with one or more dimensions, so they came up with a new technology, one that filters incoming information to the brain, thus allowing people to not go insane when visiting. One very good thing about artifacts that allow access to the higher dimensions is they work or they don't work. There are no accidents that leave body parts behind. As long as the power cells and artifacts are in working order, it is perfectly safe to enter and leave the dimensions, something that can not be said for matter transmission.
There are at least two reasons you might want to include higher dimensions in any setting. They can be used for travel as different geometries allow for access to distant locations with minimal travel. One could walk to Mars using a higher dimension (though I would limit it to much higher, such as 12th or even 20th). There may be hidden locations within the planets or Sun that can be only reached with dimension travel. The other reason is they make good places to hide people, wildlife and technology. Frozen or even active ancients may live hidden away. Cities that have no idea that the 3rd dimension civilization fell so long ago.
Unlike the extradimensional spaces I have mentioned a few times here on my blog, there are two very serious dangers to accessing the higher dimensions. As they are no more limited in size than the 3 dimensional universe, any point in them is accessable to anyone or thing. One could enter the 6th dimension and get caught up in a school of "fish" and dragged into another part of the universe (if the character isn't eaten). The other is the subtle influence between dimensions I mentioned in the second paragraph. One could exterminate all life on Earth in the 3rd dimension by accident by attracting a creature that warps gravity, activating an artifact (such as a flashlight) that emits class 10 radiation over a very large area or by moving a Moon sized body into the 3rd dimension in the same location as the planet.
Still, higher dimensions are a potential new play ground for GMs and players alike. They are very weird, can contain mind blasting aliens or artifacts and could add so much to a campaign. They may even be considered the spirit world to a people with mutations that allow them access.
