TLDR version; Teg went through something akin to the Spice Agony when tortured by a TProbe; it awakened latent physical/mental abilities that the BG knew nothing of, as well as his prescient abilities. But it was not the Weirding Way; that was just a term used to describe the sheaf of abilities and skills the BG learned and exercised via Prana Bindu training, and no real mystery to any initiate of the BG schools -- Teg was moving with the kind of speed they could only imagine.
What has happened to me?
It was something from his Atreides ancestry, he knew. Crisis had tipped him over into another dimension of human possibilities. Whatever the transformation, it was profound. He could see outward now into many necessities. And the people he had passed on his run to this alley had seemed like statues.
Odrade was quite familiar with the Weirding Way; that's not what Teg was demonstrating.. Or, rather, he used many of the skills that she knew from it, but with a degree of speed far in excess of anything the BG had ever seen.
The Weirding Way allowed the Bene Gesserit to function at pretty much the Peak of human ability, including a lot of seemingly impossible degrees of precision and speed.. But there were still limits, and Teg was functioning well past what they understood those limits to be. A BG might steal the coin from your hand without you seeing it, but the speed Teg showed was at least an order of magnitude beyond the best they could muster.
What Teg did, however, was well beyond what the Bene Gesserit could do. It's never explained in the stories, but although the BG train their bodies to an incredible level, and discipline their minds, their minds are still, fundamentally, normal. (Hence why they still needed Mentats and other services.)
Teg himself WAS a Mentat, so his mind was already organized in a different manner (as well as having some abilities/experience beyond most mentats -- see: his discussion about exploring his mind to which Duncan response "Damn few ever escape from that depth. You were lucky"), then he was tortured by a T-Probe. Reverend Mother's gain their special abilities via the 'Spice Agony', and it's been suggested that it's a 'Evolve or die' situation; that is to say that it is theoretically possible for them to develop the abilities, so the maximum torture gives them the maximum impetus to do so.. or die. Teg is placed in a parallel situation via the T-probe. And, remember, he was trained by his Mother in all the BG skills she could teach him, so he was as well prepared as any of them would have been.
He thought suddenly that the spice agony must be like this for a Reverend Mother. Surely, there could be no greater pain. He fought to remain silent but heard himself grunting, moaning. Every ability he had ever learned, Mentat and Bene Gesserit, was called into play, keeping him from forming words, from begging for surcease, from promising to tell them anything if they would only stop.
The T-Probe, as described, seems to be able to 'map' out the mind (not just the brain), and create a virtual clone of the person; shere prevents it from accessing memories, but doesn't prevent it from mapping the brain in a functional sense, allowing them to basically turn a person into a puppet.
Teg tried to open his eyes. The lids would not obey. He remembered then. They had called it a T-probe. This was no Ixian device. This was something from the Scattering. He could identify where it took over his muscles and senses. It was like another person sharing his flesh, preempting his own reactive patterns. He allowed himself to follow the workings of this machine’s intrusions. It was a hellish device! It could order him to blink, fart, gasp, shit, piss—anything. It could command his body as though he had no thinking part in his own behavior. He was relegated to the role of observer.
He understood now. The whole spectrum of his senses could be copied into this T-probe and identified, tagged for Yar to call up when needed. An organic chain of responses existed within Teg. The machine could trace those out as though it made a duplicate of him. The shere and his Mentat resistance shunted the searchers away from his memories but everything else could be copied.
Once they have him at this level, his hearing, sight and other senses should have been commandeered by the T-Probe, but Teg is a Mentat with a grasp of his own mind far greater than most people; he is able to fight it rerouting his mind and thoughts.
“He’s still hearing us!” Yar.
“Damn all Mentats!” Materly.
Then things take a turn...
The probe continued to intrude. Despite the force being applied, Teg felt his awareness matching the thing. It was learning him but he was also learning it.
Soon after, we see what you are asking about:
Mentat projection: I am moving with dangerous speed.
And his own assessment:
Mentat projection: The probe agony has lifted me to a new level of ability.
Even Teg wasn't sure what happened,tho; he speculates himself:
Try as he might, he could not resolve what had happened to him under the T-probe.
. . .
The T-probe had thrust him over a threshold into a new reality.
My take on this was that to override a computer that was mapping his brain and trying to supplant it, he was forced to organize his brain into a for more efficient configuration (or be able to shift it to that configuration at need), and was actually given the model for it by the T-Probe's own workings. A lesser man, or one lesser prepared would not have succeeded, but a combination of BG teachings, Mentat Skills and Wild Atreides genes (great for helping you make the most of a 1 in many millions chance) made him the wrong person to use the probe on; instead of it making a virtual copy of him, he was able to copy the workings of the machine, greatly enhancing his own abilities.