I'm super tired so I'm probably rambling, but I don't think it's directly describing the pickup, more like describing how a pickup makes an amp behave. It's another way of saying that (depending on the pickup and amp of course) higher output pickups need less preamp gain to push an amp into distortion. To me, a fully saturated sound is something in fuzz territory (not necessarily a fuzz tone though) where you get a lot of sustain and notes stay distorted as they trail off. (edit - Chistopher said it better) But beyond that, different characteristics become more apparent with high gain sounds, like the brash high end of a Duncan Distortion or the "grainy" quality of a 498T.
Saturation is an issue for me because a lot of bridge pickups don't have a sweet spot. Regardless of output level, when you throw them into enough gain to do the thing, they start to turn to mush. With some material that's ok, you're not really going to miss the clarity on power chords and it's fine for lead lines. You might even want a wooly tone. But if you want that clarity for arpeggiated chords with lots of notes ringing out, it can feel like playing with a blanket on your guitar, and the only way to get the clarity back is to turn down the gain, which is again ok for some material but makes for a weak sound on other parts. There's a compromise between saturation and clarity. On the other hand, my favorite bridge pickups let me dial in enough saturation to have the right feel for everything I play and enough clarity to play more complicated chords with busier rhythms without compromising. With the wrong pickups I'm constantly adjusting and tweaking my tone looking for a balance. With the right pickups, everything falls into place and I can just play.