So, if you're playing in a major key, say C major, and you play a I IV V chord progression but before you go to the V you play a bVI7 chord. So, C Major, F Major, Ab7, G Major. How come the Ab7 has such a strong pull to the V chord?
It's called "tritone substitution". The notes of Ab7 are Ab, C, Eb, Gb (or F#). This chord has two notes that are also present in the secondary dominant of G, ie D7 - C and F# - and Ab and D form a tritone (as do C and F#). Basically one can often replace a secondary dominant by the chord formed on its tritone.
It is known as tritone substitution. It works because two notes out of the four are the same as two notes from the actual dominant of the next chord played.
Let's look at the notes: G (the target chord) has D7 as its dominant. D7 contains D F♯ A and C. Its tritone sub. which is A♭7 contains A♭ C E♭ and G♭.
It's always the 3rd and 7th that become 7th and 3rd. So 3rd of D7 is F♯, 7th of A♭7 is G♭ (both enharmonic in 12tet). And - the 3rd of A♭7 is C, and the 7th of D7 is C. So the two notes kind of swapped over. That works sonically so using A♭7 instead of D7 gets us to G.
The two chords contain that tritone, which is instrumental in pushing towards consonance by being one semitone from notes in the target chord. The chords themselves (D7 and A♭7) are a tritone apart - look on the circle of fifths, they're diametrically opposite, (thus three tones from each other), so what better term to use!
The Ab7 chord is considered an Ab augmented sixth chord, spelled
[Ab C Eb F#]. The "standard" resolution of that chord would be to
[G C E G] followed by G major, but it's acceptable in modern music (i.e., 20th century and later) to just go directly to G major.
The key concept behind augmented sixth chords is that the augmented sixth (
[Ab-F#] in this case) resolves "outward" to an octave. As opposed to a dominant seventh chord (containing
[Ab-Gb]), in which the seventh (
Gb) resolves down by step (to
Theory describes dominant seventh chords as being V chords of the related key; whereas, augmented sixth chords are bVI chords in their respective key. Thus
Ab7 would be expected to resolve to
Db (major or minor); while
AbAug6 would be expected to resolve to G (by way of a
[G-C-E] chord, which some people call
I[6-4] and others call
V[6-4] or a "cadential
I remember that when I didn’t know anything about augmented 6th chords or tritone substitution and borrowed chords I was already playing and experimenting with this progression.
Now, beside the given answers I think it’s worth to recall that Ab7-G is borrowed from the parallel key C-minor, and reminds us also on the Andalusian cadence, which everybody has in the ear, whether musician or not.
Sometimes it's useful to label chords by their usage rather than by construction. Several commonly used chords have different resolutions. Obviously, the fully diminished seventh or an augmented chord can be resolved differently because of their symmetric construction. Often composers will use the same sonority (chord construction) in different ways, sometimes near in time to effect a transposition, or sometimes it just happens. The I64 (or other 64 for that matter) may be part of an arpeggiation (like in the boom-chick accompaniment pattern); it may also occur as a suspension into a V chord in a cadence (or even in a Romanesca pattern with I-V6-vi-iii6-IV-I6 being replaced by I-V6-vi-I64-IV-I6 (which is really like a passing 64).
The Augmented Sixth chords may be treated similarly. As an Augmented Sixth, these act as pre-dominant chords, leading to V or I64. One may approach the German Sixth as a dominant seventh and leave it as a G6 (or vice versa); similarly, the French Sixth can be treated as a V7b5.