Another one of those woefully misunderstood pedals/effects.
In the real world, when you pluck a string, or bash a drum or simply slam a door, the resonating object produces a fundamental or basic pitch, and also produces harmonics or multiples of that pitch. Different sound sources can be identified by the ratio or proportion of those various harmonics to the fundamental, how quickly they arrive and disappear, and other features like the inclusion of noise (e.g., with a flute), or vibrato.
What is central to ALL acoustically-produced sounds is that the harmonics line up in time with the fundamental. Indeed, it is partly the coherence and synchrony of harmonics and fundamental that allow a human being with only two ears to "hear" and separate many different sound sources being produced at once. It is as if your brain were able to sort the different sound waves striking the eardrums and sort them like cards in a deck into different suits. This harmonic goes with that drum, this one goes with that violin, and that one goes with the damn bird that won't stop chirping while I'm trying to listen to music.
When electronically produced or reproduced sound passes through a complex circuit - heck, even when it passes through a cable long enough - there is a certain amount of what is called "group delay" imposed on the signal. generally this is a result of capacitors and their charge-up time, but there are many other factors as well. The group delay reduces the coherence of the harmonic details, and their "assignability" to any sound source. You tend not to notice it quite so much when there is only one instrument playing, but when that instrument's harmonics are mixed in with the harmonics of other sound sources, the very tiny time gaps between fundamental and harmonics can create havoc with our ability to neatly organize the sound field into this sound, that one, and those other ones. The result is a bit like having a drawing of someone where the facial details were all bumped to the left or right by 1/2", relative to the outline of the face. They would sort of look the same and be roughly identifiable, but not quite. Having a drawing of a group of people like that would make it fairly difficult to recognize any single person in the crowd. Group phase delay does the same sort of thing to sound.
On top of this, there is the perennial problem of having speakers where the separate drivers for different bands are slightly different distances from the listener. Guitar speakers, being what they are - a single or multiple speaker delivering the exact same sound - don't have that problem, but when the mic'd guitar goes into a P.A. and is combined with other mic'd sources and comes out through a tri-amped system, the group delay issue arises again.
The desynchronization of fundamentals and harmonics, whether from speaker driver placement, or from capacitor-related phase delay, is what the BBE process strives to fix. It separates the signal into broad ranges and re-aligns them by imposing very very brief time delays. In fairness, no simple analog system can do this flawlessly, in terms of identifying what frequencies need to be delayed by how much, but they certainly come close enough to make a real audible difference. It won't necessarily add more treble to the signal, but it will add more audible punch and presence by more clearly "assigning" THOSE treble frequencies to THAT fundamental. It is properly classified as a psychoacoustic effect, since it does not so much shape the tone to sound different as shape the signal in anticipation of how human hearing works. I think the description that you can't tell if its on, but you notice when it gets turned off, is quite apt. A number of psychoacoustic effects work, or can be described, the same way.
These days, Barcus-Berry has licensed the process and you can buy BBE on a chip for about $3. It takes a bunch of passive components to turn that $3 chip into a fully-fledged unit, but the overall requisite circuitry has been radically miniaturized from what it started out as. You can often find BBE process on car CD decks and such.