I started off understanding but am now completely lost.
What exactly IS the theory behind using the tri-tone. How do you know WHEN it is appropriate to use a tri-tone?
In your examples, all the chords have C tri-tones. Why is that? Can i use other tritones with it?
Easy big fella; breathe.
DaNati is showing us possible uses for the C (E-Bb, in either order) tri-tone.
The "theory" behind it is: in the LH play the 3rd and the b7 (in this case the E and the Bb, respectively).
In the RH, play one of the triads he has spelled out.
I think your question is: How do I know which triad to use?
Good question. I think it's more rationale, than theory; but, I could be wrong.
For the sake of simplicity, and hopefully clarity, I'll just say this: He got the triads by playing a MAJOR 2 chord (instead of the usual minor 2 chord) of C (i.e. D Major chord); a MAJOR flatted 6 chord (i.e. Ab Major chord); a MAJOR 6 chord (i.e. A Major chord); lastly, a MINOR sharp 4 chord (i.e. F# minor).
I hope I've made this a bit easier for you to digest. Be blessed.