Sure. I will use only the triads to demonstrate, for the sake of simplicity.
For any given 7 note mode, if you leave out the root you’re left with 6 notes. Those 6 notes can be grouped in 2 groups of 3 notes each. Since we’re only discussing triads, each of those groups will be a triad. For any given mode, these triads are built on the second and third degree of the mode. Here’s an example:
Mode: C Lydian
Remaining notes after leaving out the root: D E F# G A B
Triads on the second and third degree of C Lydian are: Dmaj (D F# A) and Em(E G B)
So, the idea is, when you play the two triads in succession over the root(played, say, by the bass) you get the full modality of the specific chord-in this case C Lydian. Try it and you will see it creates a certain kind of motion and a very appealing richness of sound.
Now, you can apply the concept over a chord progression. First step is to assign a mode to each chord. Next, you look for the triads built on the second and third degree of each. Here’s a simple example:
The progression: G7-Cmaj7, a simple v-i.
Scales assigned: for G7, G Lydian b7. For Cmaj7, C Lydian.
Triads for G Lydian b7: A maj and B dim
Triads for C Lydian: D maj and E min
So, what you do is, comp over each chord using the corresponding triads in succession.
Just to wrap it up, the 3note structures that give the 6 notes of any mode, minus the root, can be of other kind, not just triads. Namely, 7th chords with no 3rd, 7th chords with no 5th, quatral chords and clusters. The book goes in great depth with all of them, both in closed and open position. But, like I said, one needs to take it slowly. I mean, the closed triads by themselves are so much fun-and not too hard.