yeast cleanup after themselves?

I keep reading about letting the yeast cleanup after themselves?
That is incredibly intriguing to me, actually. Please 'splain me what that means...

I'll give it a go in layman's terms, then one of our scientist can jump in.

After the yeast has eaten all the "easy" sugars, it turns and eats the by products (yeast waste) of their party. After they have eaten everything eatable they go to sleep on the bottom of the primary.
The end

Mighty nice of those lil critters. How long does the waste cleanup process take and is it related to airlock activity?
I joined this forum b/c I find them so helpful for learning. One day I'd like to be able to impart some of my knowledge. I wish I had the money to experiment more(and buy all the AG equipment for quality). Shoot, I get a kick out of lying down on the floor and watching the bubbles in the carboy... Hi, I'm Mark and I watch fermentation bubbles...

A lot of people here recommend leaving it in the primary for about 3-4 weeks, then either bottling/kegging or racking to a secondary. And as Revvy will tell you, it has NOTHING to do with airlock activity.

I've got an ESB that's been in the primary now for 19 days, and I haven't seen airlock activity for at least 10-12 days. I've measured SG a few times, and generally take a sip or two from the tube, and the taste has improved remarkably in the last week or so. This is my first beer that I'm leaving in the primary for so long, so I'm anxious to see how it turns out. My hopes are high!

I'm curious for the technical/scientific answer to this as well. I've heard this phrase as well, but I wonder what the actual timeline looks like.

In my (very limited) experience, the longer you leave your beer on the yeast, the more it absorbs the yeast flavor.

IMPORTANT EDIT: i just reviewed my notes and remembered that i re-used yeast in these cases where i had yeastiness, by simply re-pitching on old yeast cake -> my bad, this would seem to explain the extreme yeast flavor i've had on these beers. (over-pitching).

i would still be interested to see what is the optimum time for certain strains to sit on the yeast, if in fact that is the case.

is it not true, though, that the quicker you move the beer off the cake, the cleaner the finish?