You are just moving at what to you is normal speed, but in smaller and smaller fractions of time than the time which is being experienced by everyone around you. (Anyone who cannot slice time might, if they are really fast, be able to run a hundred metres in ten seconds. However, if I can walk a hundred metres in a second without unduly exerting myself, by the time you have run that hundred metres, I have covered a kilometre.) When you are slicing time, everything around you appears to freeze and to move to the blue end of the red shift (ask an astrophysicist). From the point of view of somebody outside the time slice, they might just glimpse you as a very fast red-coloured blur (and then put it down to a floater in the eye, or an after-image, or something).
The practical upshot of all this mucking around with time stuff is that you can get from the Hub to Ankh-Morpork on foot in a matter of days, as measured by consensus time outside your localised slice. And without exerting yourself. Sounds simple, but the History Monks spend a lifetime, or (reincarnation permitting) several lifetimes learning how to do it. Unless they're called Lobsang Ludd that is, who was born with the knack.
Ordinary mortals, even those who don't live on the Disc, are also capable of slicing time, if under great stress and if they've already been exposed to the fact that Time is more flexible than we think, for instance by exposure to Mrs Tachyon's mysterious shopping trolley.
Johnny Maxwell, when transported back to May 1941 by the aforementioned Tesco trolley, is operating under a similar burden of responsibility to that felt by Lu-Tze when he had to locate and stop the Glass Clock. He has to cover a long distance from the top of Blackdown back to Paradise Street before the German bombs start falling just after eleven o'clock, to raise the alarm that saves lives. In these heightened circumstances, he learns to slice time. Lu-Tze or Marco Soto would have signed him up as an acolyte on the spot, were they there to observe. (There is a suspicion, if you read the text of Johnny and the Bomb carefully, that Johnny was being guided by somebody or something, or the right know-how might have been given to him when he most needed it... does Roundworld have its own History Monks, and were they correcting the anomalies in time caused by Johnny and the gang leaping between 1996 and 1941? A tempting thought. Especially as on Discworld, the Abbot himself has explicitly said Do not worry about the other universes. We are there, too.)
Johnny took his grandfather with him on this time-slice. This got his grandfather a medal, which he kindly showed to Johnny one day in the Now when he was way over seventy, and not the Now when he was the young man who'd apparently sprinted so fast to raise the alarm, that the British Olympic Committee wanted to speak to him after the war.
In I Shall Wear Midnight, Eskarina Smith reveals to Tiffany Aching that she has studied time and is capable of using the phenomenon of the Travelling Now to move up and down the timeline of the Disc as she pleases. This would appear in context to be a higher-order and more specialised method of slicing time which allows the practitioner to access past and future as she pleases.
We also infer that Tiffany learns this skill too, as her older self appears to advise the younger.