If you are talking about what I think you are talking about, it’s merely using more chest voice than head voice when hitting higher notes. Chest voice is the voice you normally speak with, and resonates primarily in the (upper) chest. Head voice is a higher, thinner voice that resonates primarily in the head. The most obvious time to hear it from a non-singer is when a female mimics a choral or opera singer.
It all has to do with how the vocal apparatus works. The vocal folds are just two membranes that open and close, vibrating to produce sound. The air comes from the lungs, where it squeezes through the vocal chords when they are open, picking up the vibrations when they are closed, and passes those vibrations along to the air outside the mouth, so they can make it to your ear (or, more likely a microphone).
To produce low notes, the chords have to be fairly loose, and they overlap a bit to keep from making the air hole too large (and letting too much air through which creates a raspy sound and dries out the chords, which need to be wet to function). To produce high notes, the chords have to be fairly tight. If they overlap too much, it takes a much larger amount of air to get them vibrating properly (as the air has to be able to get out.) The larger amount of air adds pressure to the chords and makes them tire out more quickly, and can actually cause damage if overdone.
So, anyways, few non-classical male singers use the full head voice, as such a voice is very thin sounding, and takes a lot of work to make pleasant. In the 80s, the style went even further, into something closer to a controlled yelling. As for why? Probably the same macho culture that was going on elsewhere–it takes a lot of brute strength to force those high notes.
As for Astley’s characteristic voice, the other thing he did was sing with a very open mouthed style. It’s pretty easy to mimic–it’s just Elvis without the drawl. Or the voice women tend to use when they are mimicking the male voice but can’t get low enough. That style is another “macho” thing.
But I think it’s the extra air pressure that gives the impression that the singer is going to “jump up an octave.”