In part 1, chapter 7 (or, for a shorter version, episode 4), we learn the following from the false minoshiro:
It was only in the 2011th year of the Gregorian calendar that a scientist of a protohistorical civilization conclusively documented psychokinesis, hitherto thought to be a supernatural telekinetic phenomenon. [...]
Every prior experimental investigation of PK had been a disastrous failure, whether conducted in public or in a scientific facility. In 2011, however, a cognitive scientist from the Republic of Azerbaijan - Imran Ismailov - conducted an experiment in the nation's capital, Baku, that was a near-total success. At the time, the quantum-mechanical paradox in which the mere act of observing an object can induce changes in it was well known. Ismailov was the first to suggest that PK could be used to amplify this phenomenon to macroscopic levels. [...]
...The first person identified by Ismailov as possessing preternatural abilities was a 19-year-old girl by the name of Nona Mardanova. She could do nothing more than move a plastic ball, light as a feather, sealed inside a transparent tube. And yet, it is thought that it was she that caused mankind to undergo a transformation, as if she were a seed crystal in a solution, stimulating the formation of more like crystals. People from around the world followed in her footsteps, awakening a power that had 'till then lay dormant within them.
So basically, this is how it went down: some people were "inherent" psychokinetikers, like Nona Mardanova - they had abilities that normal humans didn't possess. Their abilities were, however, very limited, and not at all on the level of the people in Saki's time.
Ismailov conducted experiments on these people (specifically, on Mardanova; I don't believe he conducted any other experiments). We don't know what the experiments exactly consisted of, for two reasons: much information was lost over the thousand years that separate Ismailov from Saki; and Ismailov himself went to great lengths to ensure that nobody but he would know the full scope of the experiments.
Whatever Ismailov did, it "awakened" Mardanova. From there, it spread. We don't know how that happened either - the false minoshiro doesn't address it. It seems that once PK exists (i.e. the experiments on Mardanova happened), it spreads. Somehow. Even in Saki's time, people don't really understand how it happens - children are sometimes born without PK, despite the existence of non-PK users being an existential threat to society.