Seems less like an Easter Egg when the feature is clearly documented in the owner's manual.

I feel like a auto-tech idiot here. In all the articles I found related to the subject, I could not find a simple clarification of this feature in plain English. What the hell does this mean and why are people raving about it?

I'd describe this as a "cheat code", more than an easter egg. That is, it's a documented way to cause the system to do something that you wouldn't normally want to do, but some people might enjoy trying sometimes.

Q-B, it means that a transmission controller, one of whose design goals is to avoid spinning the tires, has a way to tell it "make a lot of noise and smoke by doing something guaranteed to spin the tires, just for the hell of it".

Q-B: Revving the engine and dropping the clutch is a way to get the engine into the part of its power curve where it generates the most torque so you launch faster from a standing start. Basically, you push the clutch in, rev the engine up to 4000 or so, hold it there, and then when the light goes green you take your foot off the pedal. This technique makes at most a few tenths of a second difference in a quarter mile, but when you're racing, every little bit helps. The M3 doesn't have a clutch pedal though (the computer controls the clutch) so normally you couldn't do this in this car. The cheat code is a way to do it anyway. It's hard on the clutch, though, which is why the car will only let you do it a limited number of times.

I think it is limited to 15 uses, then you have to gobble some glowing green orbs to get extra "peel out" points.

This could come in handy if you're parked at a lonely overlook at night with your cheerleader girlfriend and some escaped lunatic starts sneaking up on you and puts his hook on the door handle. Obviously, the engineers at BMW are a prescient bunch.