Structurally speaking, an augmented chord is the product of stacked major 3rds. So the chord G Aug (G+) uses the notes G-B-D#. I'm well aware that this chord might occur naturally in a minor (key) progression that uses the harmonic minor scale.
Yet I have seen it being used a lot on major chord progressions.
How can I explain that in harmony?
One of the more common uses of augmented triads is as a transitional chord. For example, it serves to create smooth chromatic movement from a I chord to a IV chord.
Here is an example in C Major, moving from a C major triad to an F triad in second inversion.1
X:1
T:I - Aug - IV
K:C major
L:1/1
[CEG] [CE^G] [CFA]
A similar instance comprises the beginning of the song "Brazil", but moving to vi rather than IV.2
X:1
T:I - Aug - vi
K:C major
M:4/4
L:1/8
[CEG][CE^G] [CEA]2 [CE=G][CE^G]/2[CEA]/2- [CEA]/2[CEA]/2[CEG] :|]
For the more general case of how augmented chords can be used, see piiperi Reinstate Monica's comment regarding the symmetry of the augmented chord and how by raising or lowering any of its pitches, a major or minor triad is achieved.
1Note that in the key of F major, this would be the equivalent of V - V+ - I. See Athanasius's answer for more about this progression.
2Consider that C-E-G# is enharmonically equivalent to E-G#-B#; that is, and E augmented triad. So this progression would include III+ (or, rather, V+/vi). Again, see Athanasius's answer for more, as well as piiperi Reinstate Monica's comment.
To expand slightly on Aaron's answer, the most common use of augmented triads in "classical" harmony (and harmonic languages based on that) is as a modification of a dominant chord.
That is, V-I can be modified where the fifth of the V chord rises chromatically to the third of the I chord, creating a V to V+ to I progression. Sometimes, depending on voice-leading and the particular progression, the V+ can appear without the preceding (normal major) V.
Many types of dominant chord can be used: one sometimes sees a V+ with a seventh or a V+ as a secondary dominant, etc.
That's the most standard functional use of an augmented triad, as a kind of substitute note in a dominant chord. Otherwise, as Aaron said, they mostly occur as passing chords with semitone motion.
I'm well aware that this chord might occur naturally in a minor (key) progression that uses the harmonic minor scale.
Yeah, that's more theoretical than practical. It's something that theoretically falls out of the "harmonic minor scale" which isn't really a thing used to build stuff harmonically. III in minor is almost always major. The only time III+ would ever occur in standard "classical" progressions would be if the chord was actually V+/VI, i.e., a secondary dominant.
(FYI: A bit of a digression, but the "harmonic minor scale" is a theoretical construct originally used as a shortcut to find the most common notes for most chords in minor. But it wasn't originally intended as a practical reflection of the nuances of minor-key harmony. In the 20th century and in some folk musics, it came to be used as an actual scale as the basis for melodic and sometimes harmonic creativity. But III+ is simply not a "normal" chord in the minor mode, at least the traditional use of minor-key harmony. One could just as legitimately build an augmented triad on most other scale degrees.)
I get the impression that you're a bit unhappy about using a chord that isn't diatonic. You'd like it to fit some scale.
Think of it more as a modification of a diatonic chord. In a G7 - C progression, it's common to use G7+ (G, B, D♯, F) instead. D, the 5th of the chord is transformed from an almost inactive fill-in tone (remember the 'you can omit the 5th' rule?) to a very active one! It had darn well BETTER resolve up to E!
An augmented 5th can be used to intensify any dominant-function chord in this way.
Another common use of the augmented chord is tonic, raise the 5th, raise it again, come back again. Three flavours of the tonic rather than anything functional. Classic example 'Maybe This Time' from 'Cabaret'.
As an augmented chord has a raised fifth step, one good use is to follow the augmented chord with a chord using the sixth step. (Aaron's examples show this.) In general, chromatically raised notes tend to be resolved by moving up a half-step.
Until the late 1800s, the augmented chord on scale step 3 in the minor mode was rarely used. It was a matter of style. The mutable steps in the minor were mostly used to help create an authentic cadence V-i. Of course, the Eb-G-B augmented chord (in C-minor) could easily lead to an Ab chord.
