I think I found out how psionics works. Again, as with my Arcane magic theory (appearing in my Advanced Beings with Alternate Energy Sources post and summarized in this post earlier), I am assuming that similar types of magic work under the same rules of reality from world to world. I drew particularly from Athas and Eberron when making this theory, as these are the two worlds that are the most psionicaly developed, and thus are the worlds were there is sufficient fluff available to explain the crunch. So here is my theory.
According to the Dragonshards article Psionics in Eberron by Keith Baker, the the fate of the planes of existance in the eberron cosmology are dependent upon the fate of the prime material. We know that this is very true in eberron, as it is directly stated in regards to the current state of the elemental and paraelemental planes and the current necessity for elemental priests on Athas, as well as the deplorable state of Athas reflected in similar states in the Grey, and presumably the Astral. The presence of the planes also, obviously, affects the material plane in turn. When the planes are in perfect balance (as with the Great Wheel Supercosmology), this produces a natural and easily manipulated energy flow which can be diverted to use in spellcasting without detracting from the overall flow. Obviously, Athas is not one of those worlds where this is an option.
However, another notable affect of the planes in the article was that their energy could be "channeled and focused through the lens of the human mind", producing psionics. It is not entirely clear what is meant by this, but it is implied that, at least in Eberron, the energy of psionics comes from one of the 13 outer planes (typically Dal Quor the Region of Dreams or Xoriat the Realm of Madness).
This is where Athasian lore is particularly useful. In Athas, the core assumption is maintained that psionics is linked to the Astral Plane. Further, it is stated that within the Astral there are a number of temporary demiplanes called mindscapes produced by manifesters who are in the midst of using a psionic power. To borrow from David Schwartz's Dark Sun Cosmology page (found here: http://siltskimmer.org/_access/dsplanes.pdf): "A Mindscape is the psychic construct that links the mind of an individual to his body. Mindscapes are normally unseen, except during psychic combat. Each mindscape appears as a location of importance to it’s ‘owner’." While the page later says it is found in the Grey, I assume this to be an error, as it is contradicted in his portion about the Astral when it says the mindscapes are found there.
This implies that whenever a psion manifests a power, he is projecting his mind into the astral, creating a small bubble of space, and forcing his will on the resulting demiplane. This makes sense, because it fits what we already know about the Astral: it is heavily malleable to thought.
Now, for how this affects the world, we can borrow from Eberron. In eberron, the walls between the planes are often very thin. Areas like this are called manifest zones, and can be enhanced by spells or psionics. If psionic powers in Athas produce a temporary manifest zone to the manifester's mindscape, he could manipulate the area as expected.
So where does that leave Eberron? Well, I would propose that this process works with any plane with similar morphic traits to the Astral Plane. Thus, since Xoriat and Dal Quor are morphic, they would be natural sources of psionics, especially given that the outer planes of Eberron are naturally inclined to produce manifest zones. In fact, Dal Quor already has a natural equivelant to mindscapes, the dreamscapes, which is where the minds of dreaming mortals go. Psionic natives simply have the ability to force there will on their own plane and then project it into the material.
In Krynn, psionics is impossible because the gods don't like it (an explanation that actually makes sense in a world of heavy-handed gods). In Dread, psionics involves forcing one's will on the mists themselves, but has the side affect of potentially attraction the attention of the Illithid God-Brain and sucking the user into the Bluetspur.
As for how the powers themselves are actually produced, it's reletively simple. Whenever characters move under their own mental power in the Astral Plane, they are using a form of psionics. As such, if a character is using most powers, he is simply manipulating his personal reality. For more refined powers, such as telepathy, one could imagine waves being sent out through the medium of the demiplane. In Athas, powers of this sort may produce a psionic display as the waves are produced; indeed, if this explanation is correct, psionic powers on any world should produce displays.
That's all folks. Musings from a Zik-Chil on a train.