Bach was exceptionally good at writing parallel lines of music while keeping them separate and individually interesting.
The technical name for this is counterpoint. While counterpoint never quite died out - it's still used in pop and film scores - no one else has ever written counterpoint quite as dense and inventive.
And Bach didn't just write it, he improvised it, with six or even eight lines at a time.
WTC is an example of writing in different keys - identical note scales that start on different root notes.
Before modern tunings keyboards were tuned to a specific key. The other possible keys were always more or less out of tune.
With modern tunings the sourness was smoothed out. None of the keys are exactly in tune, but they're all similarly good-enough.
Bach's tuning probably wasn't quite the same as a modern tuning. But it was close enough to allow him to write a book of pieces that worked through all the keys on a single instrument without retuning - which was a novelty for the time.
WTC is also a kind of pattern book of contrapuntal and compositional techniques. So it made a huge impression - and still does.
If you're not a musician I wouldn't get too distracted by the theory. It helps to know what a fugue is and maybe a little about form, but the music is very good at speaking for itself as a self-contained experience without insisting that you understand how it was put together.
When I looked for it now, they seem to have something entirely different in its place [1]. Incidentally, they also have a talk on Bach's Preludes and Fugues [2]. I don't know if these are as good as the one I heard years ago. I'll find out tonight.
I have read a very little bit from music theory textbooks. I don't play an instrument, so the books quickly became tedious for me.
I thought it should also be noted: species counterpoint was being formalized just around Bach's time by Johann Fux[4], and modern clarifications and extensions to Baroque formal species counterpoint are largely derived from reverse-inferring rules out of Bach's masterful use of counterpoint and his inventive ways of twisting the rules (often to make his own consistent contrapuntal language). In this sense, WTC is an absolute masterclass of the rules that formed the core of the defining era of European music, which is why it's so famous.
[1] http://tobyrush.com/theorypages/index.html
[2] http://www.ars-nova.com/CounterpointStudy/intro.html
[3] http://www.listeningarts.com/music/general_theory/species/me...
I took away a lifelong lesson from this 1 hour listening session, and all it required was the score and my attention. Counterpoint is a rhetorical tool, and I think it can be fundamentally understood and appreciated even without performing the full dissection.
I also recommend Rachmaninoff's Etude-Tableau Op. 39 No. 4 in B Minor for counterpoint appreciation. That piece never fails to blow my mind. https://www.youtube.com/watch?v=hWLotW5AKjg