The augmented 6th chords were harmonic explanaitions in classical theory to explain temporary dominants that approach a chord from a half step above. Today we refer to them as "tri-tone substitutes."
If you went to the web site that was directed in the above post, each of the augmented 6th chords had as their outside notes F - D# which is the interval of an augmented 6th. If you respell those two notes then you have F - Eb a min7 which is the root and 7 of an F7 chord. As you will not, each of these progressions went from the augmented 6 to the V. The respelling from Eb to D# made that note a temporary leading tone of the V chord and indicated that the root of these chords was actually the D# and that they were actually being used in first inversion which allowed them to behave according to the rules of tonality.
Today, we simply don't worry quite so much about this harmonic correctness and consider the chord to be a bVI (or temporary bVI) in a given key so that we think of the progression as bVI - V - I.
If you also look carefully, the Italian 6 in F,C, D# is simply an F7 without its 5th (C) The german is the F7 with the 5th, The french is F7 b5 and the doubly augmented an F7 #5.
The tri-tone substitute idea is that The V of the E7 chord in the example is B. The tri-tone substitue is the chord with its root a tri-tone (+4/dim5) from the root - in this case F. Then build your chords from there. Today we are not always limited to dominant type chords. In many cases a Maj7 or even a min7 will give an interesting sound.
You can actually use a tri-tone substitute for just about any part of a chord progression and get some interesting results.
For a common I VI II V I progression C Am Dm G7 C try, I bIII bVI V I (C Eb7 Ab7 G7 C) the Eb is a tri-tone from A, the Ab a tritone from D.
I hoped this helped a little.
