I asked my teacher about how I would apply these, and he gave me a great explanation and a demonstration (which was amazing!).
He's not just asking me to memorize modes. I started out doing this, and was unsuccessful. He told me he wanted me to use my ears and listen to them. To learn proper fingering and economy of motion on the fretboard, and to play the modes up and down the entire fretboard and across all strings (two full octaves on my 5er).
I can see where he's going with it, and I can see how this will really open things up for me.
I took lessons from a guy Adam may know - and his teaching style sounds very similar when you talk about modes.
I went into my lessons somewhat mystified by 'the modes' and like many, felt that learning them was critical to overcoming what I thought was an obstacle to my ability to express myself more musically. There is value in learning the modes and practicing them - and ultimately the most 'applicable' result for me was essentially 'knowing where all the good notes are all over the neck - no matter what'. The modes simply became a 'road map' to every single good note on the neck.
While learning about them didn't turn me into a better player, not knowing them was getting in my way of thinking I could be a better player. So in a way, learning them did allow me to become better - but it's not because of what they are - it's because I no longer think of them as some sort of 'lack' in my tool kit.
After practicing all the modes scales and arpeggios, 2-octaves, all the way up and down the neck, all the "good" notes were exposed. I put "good" in quotes because, as I am sure Adam will teach you, there are no such thing as good or bad notes - and playing 'out' is as important as playing all the 'in' notes when it comes to being musical.
One thing that helps demystify the modes is seeing exactly how they 'work' in relation to each other. For my own benefit and for fun, I started writing a program that generates all chords for all modes - for no other reason than I could see the rather mathematical relationship in the interval sequences and was interested in exactly how much I could leverage that to expose the math in music. Here's my simple mode object that lists out the intervals of each mode (rewritten for clarity of display):
[2,2,1,2,2,2,1] = ionian
[2,1,2,2,2,1,2] = dorian
[1,2,2,2,1,2,2] = phrygian
[2,2,2,1,2,2,1] = lydian
[2,2,1,2,2,1,2] = mixolydian
[2,1,2,2,1,2,2] = aeolian
[1,2,2,1,2,2,2] = locrian
And a version to highlight the obvious patters;
I highlighted the last interval in the first mode to show how it moves from mode to mode. Look at all the rest of the intervals and you'll see what each mode 'means' to the other with regard to how the interval groupings move from one mode to the next.
[2,2,1,2,2,2,1] = ionian
[2,1,2,2,2,1,2] = dorian
[1,2,2,2,1,2,2] = phrygian
[2,2,2,1,2,2,1] = lydian
[2,2,1,2,2,1,2] = mixolydian
[2,1,2,2,1,2,2] = aeolian
[1,2,2,1,2,2,2] = locrian
And another:
[2,2,1,2,2,2,1] = ionian
[2,1,2,2,2,1,2] = dorian
[1,2,2,2,1,2,2] = phrygian
[2,2,2,1,2,2,1] = lydian
[2,2,1,2,2,1,2] = mixolydian
[2,1,2,2,1,2,2] = aeolian
[1,2,2,1,2,2,2] = locrian
...and another (it should all be pretty obvious at this point):
[2,2,1,2,2,2,1] = ionian
[2,1,2,2,2,1,2] = dorian
[1,2,2,2,1,2,2] = phrygian
[2,2,2,1,2,2,1] = lydian
[2,2,1,2,2,1,2] = mixolydian
[2,1,2,2,1,2,2] = aeolian
[1,2,2,1,2,2,2] = locrian
This really doesn't mean anything, musically - but it's definitely a pattern that exposes the mechanics behind the intervals as they relate to each other in sequence. I found that understanding this helped to explain 'what' the modes are in a very cold, analytical sense.
Now the modes do have "emotional" implications (moods) that come from the mathematical facts surrounding these intervallic relationships that can be applied - but that's more of the art side and less of the science side. But from a strictly mechanical perspective, they're just shifting the intervallic relationships as you traverse the modes OR playing the exact same intervals starting at a different step each time.