Here's an explanation of John Coltrane's "Giant Steps."
Tommy Flanagan, the pianist who struggled a bit on the original Coltrane recording, put out a recording of Giant Steps with his own trio in the 1980s. Having had enough time by then to woodshed the changes, his improvs are exquisite. https://www.youtube.com/watch?v=O3PWynH-CEc
The point is that this Vox presentation and many other articles on the Coltrane changes present them as so revolutionary that improvisers struggle, but on its own that's not true.
There are records which predate Giant Steps which are equally harmonically interesting and challenging and which similarly modulate through many key centers, like Con Alma [1] by Dizzy Gillespie from 1954 or Lush Life [2] by Billy Strayhorn from 1948.
What Coltrane did with Giant Steps was akin to calling for Lush Life at 300 beats per minute.
On more straightforward changes, I can usually internalize the melody and harmonic motion well enough to form a sensible solo, but I've got my limits. Fortunately the era of "Giant Steps" getting called on every gig has more or less passed.
But I acknowledge that in that context, it's an "out there" harmonic adventure which builds tension, to be released when the final section harps on the tonic.
I still think that it's mostly the tempo that makes Giant Steps feel alien. Modulating through major keys by major thirds isn't that exotic, but doing it so quickly that ear just doesn't get a chance to orient itself is.
IMO it's not only the tempo. It's also the fact those changes kind of subvert any feeling of tonality, so you can't really rely on tonal formulas (i.e. cadences) when improvising, so you need to come up with other ways to structure your improv, while constantly shifting between modes/scales/patterns. Becoming fluent in this kind of context is a huge challenge, even in slow tempos.
For instance consider that pitch class inversion of a major scale around its root results in precisely a major third key change. E.g. if we invert C major, we get C Phrygian, and C Phrygian is a mode of A♭ major: Coltrane change!
Try improvising around the two chord loop CM7, C♯M7 (or D♭M7), CM7, C#M7 by shifting between C major and its inversion C Phrygian; it is very melodic and intuitive.
You can feel/invoke similar mood changes when following the Coltrane changes.
C#M7/D♭M7 contains a Fm triad, so it evokes hint of the common parallel minor device. C Phrygian and C Aeolian/minor are in fact only one note off, Phrygian is Aeolian with the second degree flattened.
A Coltrane key change can be fibbed by the improviser by a parallel major/minor shift which is so ingrained into the pop music repertoire that it's easy to find familiar idioms that the listener will latch onto. You omit one note that doesn't work and are good to go.
Here is a cool thing about the Coltrane keys (related to the above semitone chord change). In each key you can pick a major triad such that their roots form a semitone progression. WE already identified C,C♯/D♭ from C and A♭. Then the third key, E, gives us the B triad.
The B triad in E major is at the root of the Mixolydian mode. The C triad in C is of course at the root of Ionian. The D♭ triad is at the root of Lydian. I.e. we have the I chord from one key, IV chord from another, and V chord from the third, and they end up chromatic. This can be exploited in improvisation. As the accompanist goes through changes you can just move by a semitone and then change one note in the scale at the same time. E.g. if we go from Mixolydian to Ionian, we sharpen the 7th. At the same time we go up a semitone, since the root is moving from B to C. Then for the next change, we do it again; root moves to D♭, scale shape changes one note to make Lydian. You can emphasize the chromaticism so the half note shift intent is clearly conveyed, while the changing note in the scale takes the sharp corners off it and makes it more interesting. The small chromatic movement helps the listener make sense of the weird chords that are going on (because the Coltrane chord progressions have a II-V-I in each of the three keys and whatnot).
Speaking of the II-V-I progressions in those keys: here is one more thing. It's possible to see a tritone movement there that evokes the tritone substitute.
Here is why. Consider the II-V-I in the key of C, where I is C. Now consider II-V-I in the key of E. What is the II note? It is G♭. That is exactly a tritone away from C.
So it's like we have a circle of fourths with tritones thrown in: D-G-C-G♭-E♭-A♭-... 4, 4, ♯4, 4, 4, ♯4, .. (if we do the key changes in that direction).
I use this as a trick to find the chords on the guitar fretboard.
The tritone has musical implications though, too. Suppose we alter the CM7 landing note to C7 and also change the G♭m7 II chord of E to G♭7. Then we have a note-sharing tritone root movement between two dominant chords.