I also have an easier way to explain Diavolo's ability since people still don't understand:
Don't think about the entire A-B-C or whatever nonsense, it's too complicated when it can be explained in an easier way and with an analogy.
Have you ever heard of a
lag switch? That in essence is EXACTLY what his ability is. Just like in an online shooter, the lag switcher can freely move around and from his perspective, everyone else can too. The problem? From everyone elses perspective, they're stuttering and locked in the same general place, confused as to how their inputs suddenly go through all at once only for them to immediately be killed by the lag switcher who doesn't share that problem.
That's the analogy, the simple explanation:
Once the ability activates, Diavolo can see all actions (humans or otherwise) from the start to the end of the "time skip"
superimposed onto the reality around him. If you shot a bullet and he activated it, he could see the end "frame" where the bullet is supposed to land in his chest.
The catch? Whilst everyone else is forced to move in a predetermined way in accordance to their afforementioned end "frame", Diavolo can adjust his fate and move however he wants, putting himself in a better position. All memories during this period of time are erased, leaving people confused as to how they got to where they are (but not Diavolo, he keeps his).
Important to note he doesn't actually kill anyone or touch them during this period, but only after.