I haven't gone through this entire course yet, but I did go to college for Trombone and Piano before diving into the guitar, so I can help to answer some of your questions.
The term "Diatonic Chords" refers to the set of chords derived from the scale/key you're using. For example, let's look at the C Major scale (it has no sharps/flats so it's easy to see/demonstrate theory concepts). The first chord (often indicated with a roman numeral I) is made of the notes C-E-G, the ii chord (lower case roman numeral to indicate a minor chord) is D-F-A, iii chord E-G-B, IV chord F-A-C, V chord G-B-D, vi chord A-C-E, and the vii° (diminished chord) B-D-F.
This information is useful because it gives you a shorthand to learning songs. Most songs will stick strictly to the diatonic chords of its key. And out of the songs that don't stick to the original key, most only go as far as to use chords that are diatonic to closely related keys. For example, a song in C Major might throw in a chord diatonic to C Minor. If your band member just wrote a new song and you both know the diatonic chords for the key it's in, you can quickly explain the chord progression by simply saying "I-V-vi-IV" or "I-vi-ii-V". In addition, knowing the diatonic chords for multiple keys will allow you to change a song to a different key. Let's say you want to play a song in the key of G, but your vocal range fits better in E. Knowing both sets of diatonic chords allows you to quickly play the song in a key that's more comfortable for you to sing in.
As far as applying the knowledge, perhaps you could try combining diatonic chords in different ways and see if you can find some familiar-sounding chord progressions (Hint: the two I've listed above are used in an extremely wide variety of songs). I would also highly recommend becoming familiar with the diatonic chords of the Mixolydian mode, the Rock genre is absolutely SATURATED with songs in the Mixolydian mode.
Does this help? I'm an absolute NERD for music theory, so I'll be happy to answer any other questions.