How psionics are handled IMTU requires a bit of background, so bear with me. I don't use a lot of the more 'out there' powers like teleportation or even pyrokinesis and so on.
I generally run psionics in a way that isn't too far from how I see the canon TU to work, just with more rules behind the scenes to explain how they work.
Psionics are not a natural thing; they were constructed. Grandfather didn't really even invent them, he adapted something he found in the records of a still more ancient race, one that was fascinated by the construction of "subspaces" - essentially manufactured alternate dimensions. They used it to "escape" the inevitable Heat Death of the universe, or rather, they rendered the Heat Death irrelevant by making subspaces where the passage of time was vastly different from "our" universe to the point where they could stretch the time between seconds to essentially infinity, making the Heat Death irrelevant as they had expanded the period between the big bang and heat death into a bounded, but infinite space.
Grandfather of course became fascinated by the creation of subspaces as well. They allowed him to expand his consciousness and perform many tasks simultaneously, far beyond the number of limbs and sensors on his "meat" body. However, even then he ran into limits, and then moved to having children and whole thing we're all familiar with in Traveller canon.
Psionics are, therefore, Grandfather's "work space" and what we think of as "psionic powers" are actually "machines" (using the term VERY loosely) which this subspace which are "attached" to the psions and actually receive instructions from the psion's brain (the rules of this subspace are different - and some kinds of radiation "resonate" within that subspace without actually crossing through). Psionics were very handy for Grandfather, allowing him to do things like seal the doorknob of a door within the door to prevent non-psions from opening the door.
Although Grandfather only granted the ability to use psionic subspace machines to his children, his children also had plans and projects of their own, and they eventually designed psionic machines for humans and Vargr to allow their "trustees" to operate Ancient facilities in their absence (and Grandfather's children were increasingly absent, something about Grandfather wanting to kill them).
To limit the use of psionics to trustees, Grandfather's children essentially made psionic machines only respond to people with certain genetic sequence. As the War wore on, the children realized they'd be away for longer and longer periods of time, and so it became too much of a bother to continually keep coming back to grant new trustees powers after the old ones died, so they scaled back the security measures to only look at some parts of the genetic code of a user, allowing many descendants of a psionic power user to use them as well. That, and the fact that one of the machines is the "copier" used to copy machines from one user to another (what modern psions think of as "learning" a power is actually the teacher copying his or her machine to the student and the student learning to use it) was seen as sufficient protection against too many psions.
Of course, the War didn't quite end the way the children of Grandfather intended. While many of these psions were killed in the war, many more survived and had children. These children are ancestors of psions in the present of the campaign.
Nobody (not even the Droyne) know about how this stuff works; you can jig up something to "see" psionic machines on someone even at TL12 -- provided you understand the theory, otherwise people will discover it around TL20. So nobody really understands how psionics have these powers but their genetics aren't different from a non-psion's. If you could see it, psionic powers are sort of like "tentacles" or "cybernetic limbs" (again using the analogies very loosely) sprouting from a person's head; a telepath literally just extends a "thought reader" machine over to the person's whose mind he or she wants to read it thrusts into the person's head (non-obtrusively in our universe) and reads the faint electrochemical traces of a person's thoughts; this presents a hard limit to the range and power of psionics, all constrained by the physics of the psionic subspace which are immutable (but different from our own).
It's also unclear to psionic researchers why psionics seem to roughly run in a family, but often many children won't be able to use powers (breeding introduces new genes which may produce the wrong combo of dna to trigger the "user database" or they might need someone else with machines that happen to open to that sequence - a pretty slim chance); this trait is referred to as paragenetic inheritance.
Because you have to think a certain way to use psionic powers (to "program" the machines), getting training when you're younger makes you a stronger psion. As you get older your brain becomes set in its way, so it becomes increasingly harder to learn to be a psion even if you could use the machines.
Pure Solomani have a vanishingly rare chance of being psionics (thus why there were never verifiable psionics on Earth) - Earth people are the ones who weren't taken after all, and thus Grandfather's children never granted them psionics. Nevertheless, with the wide variety of human varities on Earth, there's a very small chance someone might just happen to meet someone with psionic powers that can be unlocked through sheer chance of genetic inheritance. While the Zhodani have an incidence of around 35%. Other races have a very low incidence of psionics.
This low incidence (usually less than 1:10 000 people), in addition to the fact that most "potential" users never even bother seeking out training, and that you need to find a teacher with the machines that will be unlocked by your DNA make psions pretty uncommon. Psionics institutes are aware of paragenetic inheritance (since the Zhodani have figured it out) so they usually ask a someone who wants to be tested about their geneology, and try and someone who is as closely related to the candidate as possible to do the testing, failing that, institutes have multiple people test a candidate; many hopefuls spend their life savings and years of their life going from place to place trying to get tested by different teachers. The most successful psionics are trained by relatives and other family members. Most psionic institutes are run by a family or a clan. Similarly, it makes it easier to monitor and supress psionics by the Imperium for the same reason.
