You're not the only one who doesn't get it! I'm obviously one of the students who has Lydian ears so to speak; I just felt it from the start.
I think what I like about it is the same feeling of not being tethered to chordal roots I was talking about in the other thread; this tendency towards a kind of literalism we all get when we see a chart where Cm7 is automatically putting us in mind of some kind of C minor mode or else Cmin7 arpeggios with extensions or neighbour tones.

When you look at it Cm7 F7 interpreted as a member of Eb Lydian scale or some more dissonant variant like Lydian augmented or Lydian diminished encourages you to use the essential guide tones of the chord. If I play an Eb centred line encompassing the fifth (Bb#) the flat 5 (A) and the root itself, I've basically covered the essential chord tones staying clear of the roots and fifth which is the bass player's territory anyway.

From that standpoint alone I feel it's already a successful approach and I haven't really begun to scratch the surface of it's deeper implications. Basically it's guiding you towards thinking of potentially using all 12 tones within any harmonic situation at any time. Perhaps you have to be Eric Dolphy, Miles, Coltrane or Russell himself to really make it work but listening to their use of it certainly inspires me to give it a try!.

Alan