I use this sequence of chords, Dmaj7, C#mi7, Emi7, E7, Eaug, Amaj7(A C# G# E), A#5#3 (A F A D), Amaj7 as an introduction for songs "Try a Little Tenderness" and "No, You Don't Know Me." Why does this progression sound acceptable even though it is nowhere in the body of the song (in the key of A)?
TL;DR: The chord progression clearly establishes the key of A major, which is why it works with A major songs even though the progression itself is not in the song.
It helps to look at the end of the progression first. The chord labeled A#5#3 is really just Dmin. So the final three chords are:
Amaj7 Dmin AMaj7 (Imaj7 iv Imaj7), which is a perfectly standard plagal motion establishing the key of A major, which is where the body of the tune lies. A very smooth introduction.
Working back, consider the E7 Eaug Amaj7. This, too, is a very common progression. Essentially it's just V-I, but with the augmented chord interpolated in between, creating chromatic voice leading from the E7 chord's B, through the augmented chord's B#, and then to the A chord's C#.
Moving to the C#min7 Emin7 E7, consider the progression instead as E6 Emin7 E7. Now one can see that E6 and E7 are simply one note different: C# moving to D. The intervening Emin7 chord again provides some chromatic interest by moving from G# to G and then back to G#.
Finally, the beginning of the progression is just IV7 iii7 in A major, which is, again, not so unusual.
So, in short, the entire progression really comes down to an expanded IV V I sequence with some chromatic voice leading between the primary chords. The progression clearly establishes the key of A major, which is where the tune is. The same progression could be transposed to provide an introduction to other songs as well.