So regular LCDs just have one big backlight that covers the whole of the screen (so blacks appear a washed-out grey, because they're also being backlit), but Mini-LED backlighting has lots of different zones (think it's something huge like 10,000 zones on the MBPs?) where each has a separate backlight. so these can be switched on or off independently, giving you pure blacks in some areas even when other zones are turned on. One downside of this is something called “blooming”, where sometimes black areas have a kind of light halo bleeding into them, because the zone needs to be illuminated to show (for instance) a white portion of the image, so the black area surrounding it gets illuminated as well. This is different to OLED and MicroLED, where the LEDs themselves provide the luminance of the screen (so no blooming) — in Mini-LED the various backlights provides the luminance, whereas the LEDs in front of the backlight tell it what colour to be.
ProMotion is just to do with how often the screen refreshes — on the MBPs I think it's between 10–120 Hz (times per second), so if what's on screen is static then it'll refresh less often, and if you're scrolling it'll ramp up to 120 Hz to make it look smoother. And it'll play video in its native frame rate, e.g. 24 Hz, which both saves battery life (because the screen needs to refresh less often) and gets rid of any judder that might've occurred from trying to play a 24 Hz video at 60 Hz (which most displays refresh at).
I'm not an expert in this stuff so might've got one or two things above slightly wrong, but the gist should be right.