Further disclosure: @ohnobinki now completely gives up on hoping to understand this so-called turbo mode at all. You may visit him in the psych ward if you dare.
Partial disclosure: the following information about how the turbo button works is more correct than it used to be but fails to make sense in connecting the turbo toggle switch found on old PCs to the term turbo coffee.
Back in the day, before the modern concept of overclocking, there existed i386 computers with turbo buttons. Now, exactly what the turbo button did when it physically appears enabled varies between computer models. But let us start with the 8086, and ancient processor that ran a bit slower than 5MHz. Later, a turbo edition of the processor which ran at 8MHz was released. As time passed, manufacturers continued improving the CPU speed. However, a lot of old software was written to be dependent on hardware timing, meaning that it would only function if the system ran at the turbo speed of 8MHz. For example, some ancient DOS games would count CPU cycles with the assumption that the system is running at 8MHz instead of using a proper timer or detecting the clock speed at runtime to control the speed of animations or AI. Running such a game on a faster CPU would result in the user immediately losing the game without even being able to react unless if it were run at the original turbo speed of 8MHz. Thus, the primary feature of the turbo button was to toggle between supporting legacy and modern better-designed software. Yes, it was a kludge to compensate for bad programing (in hindsight).
Of course, when using software that crunches numbers or is better-written, the slower turbo speed cripples performance. However, originally, turbo, referring to an 8086 chip with double (8Mhz) the original (4Mhz) speed had the positive meaning of being speedier. Thus, the turbo option on i286/i386 PCs was for people with the retro/beige spirit who wanted to hang on to the original sense of awesomeness of the 8086. I propose that turbo coffee refers to coffee that is twice as good as other coffee. And, to come back to the subject at hand, toggling turbo to return to the slow 8MHzish turbo speed (which, in a retro sense, remember, is twice the speed as the original 4MHz 8086) is like giving a despondant human good old extra strong coffee. Compared to the original 8086 or original coffee, you end up with faster context switching, lower latency, and an increased alertness/jumpiness.
It is readily obvious from this single picture that, without strong coffee, Nate Musch would have been quite glum and slow, just like a the original 4MHz non-turbo 8086. And, well, when a bunch of caffeine-appreciative retro-computing geeks gather together, such as in #retrobox, and one member braindumps his thoughts on whether or not to get another cup of coffee, and another community has already invented the term turbo coffee, you get ridiculousness. You get this website.
This site is run and maintained by Nathan Phillip Brink (binki) / Oh! No! Publishing. That probably explains why it uses the XML serialization of HTML5.