The Augmented 7th chord, otherwise known as 7#5, is really a dominant 7th chord with a raised 5th scale degree. That's the simplest way, or you can think of it as an augmented triad with a flat seventh.
For example, C7#5 would be spelled.. C E G# Bb and would resolve to a F or B chord.
The 7#5 chord gives you a stronger sense of drama and urgency than a plain dominant7 chord. The chord still resolves down the same as a dom7: down by a P5 or half step (tritone sub)
In the dom7 to I chord progression (ex. G7 to Cmaj) the b7 resolves down a half step to the third of the I chord. In the 7#5, the #5 resolves UP a half step to the third. (ex G7#5 to Cmaj - the note D# resolves up a half step to E. So you have another chromatic approach to the third of the I chord.
Remember once you sharp the "5" (fifth scale degree), you can no longer play the regular fifth scale degree or it will clash. It is no longer "legal" in the traditional sense.