None of my players had foul trouble all match long, so it is irrelevant whether I had Let Them Play or Sit Them, isn't it.
My observation is that Let Them Play results in less subs regardless of foul trouble. My guess is that in certain situations, if the coach cares about fouls, he might pull a certain player earlier in order for him _not_ to pick up fouls (in addition to immediately benching them when they pick up a foul).
For example, the coach will pull your starter with 2 fouls in the middle of the second quarter, so that he doesn't pick up his third (since picking up a third results in benching him immediately until the middle of the third quarter, for example). Let Them Play seems to completely ignore these considerations.
I have never really done any testing with this, so the evidence is anecdotal.
I know they do conserve energy, but what do they do it that much?
No idea, up to very recently I thought they didn't at all.
Have in mind that players regenerate energy in timeouts and at half-time.
"I don't know half of you half as well as I should like; and I like less than half of you half as well as you deserve."