I don't want to speculate on how representative that lead sheet might be of the original song, but the question about the dim7 chords in the posted chord progression can be answered regardless.
Dim7 chords are symmetric and there are many plausible explanations for why they "work" or what they do. One way is to not think about it, just play the dim7 in different songs and different contexts, and you'll learn how it works naturally. Another way is to see them as dominant chords, and because of the dim7 chord's symmetric nature, each dim7 can be seen as four different dominants. See this answer about how it works: Why don't diminished 7 passing chords work going downhill?
For the G#dim7 -> Am7 step this is clear: G# is the third of E7 and G#dim7 works as dominant for A-based chords. Or you can think of the G# as a leading note to A.
Why does the D#dim7 -> C/E step work? The D#dim7 doesn't work as a G7, so it's not a dominant for C? No, but it could work as a B7, which is a dominant for E ... so, bass goes to E, but the rest of the harmony goes to C?
Like I said, you could just as well play the song from the lead sheet, maybe play it in different keys, and learn the D#dim7 -> C/E step as an atomic basic ingredient that doesn't need any further explanations. Just like you probably don't need a theoretical explanation for why subdominant IV chords sound like they do. You just use them and that's it. :)
For me personally, one such "atomic" progression is C - Cdim7 - Dm7 - G7. Why does that work going to Dm? Maybe because there's the downwards motion of the minor third E-G -> D#-F# - > D-F ? But is that a reason? Can you generalize that and use it in other contexts?