As Shufflebeat says, these chords are often used a passing chords. They originated in the use of melodic passing notes over simple triads and then became incorporated into standalone chords.

Eg. In going from E to A the melody or an inner part might pass from B to B sharp to C sharp. This passing note creates an augmented chord. The augmented chord behaves like a seventh chord, except that the augmented note pushes upwards towards a resolution where the seventh note would push downwards. The dissonace in these chords creates tension and their resolution a relaxation, so overall there is a movement and purpose in the parts, a continual process of creating and resolving tension. This can turn a succession of notes that nobody cares about into a gripping narrative. Or that's the idea.

A common use of the augmented chord is at the start of some rock 'n roll songs. Creates tension and pushes the listener towards the song. 'Riding along in my automobile....'

One handy thing about dim 7th chords is that it consists of four notes, where dropping any one of these notes a semitone creates a dominant 7th chord. This means it is easy to make radical but strong modulations in four very different directions from a dim 7th chord.

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