What was the purpose or reason for it appearing to begin with? Were words with double-ss needing some typographic distinguishing? What warranted a new character that broke through the normal height of lower case letters, when regular s was doing just fine? And why only one of the S's in a pair? Puzzling.
The German language allows you to compose several words into one long word, which is no longer structured by spaces. This can lead to ambiguities, especially where a sub-word ends with an S. Due to the rule that words always end with a round S, these ambiguities were resolved.
Examples from German wikipedia: https://de.wikipedia.org/wiki/Langes_s
Kreiſchen (Krei·schen, shouting) vs Kreischen (Kreis·chen, a small circle)
Verſendung (Ver·sen·dung, posting) und Versendung (Vers·en·dung, end of a verse)
It entered usage because we needed a lower case S when Carolingian monks first developed cased letters in the eighth century, and the Romans already had a similar shape (the "medial S"). For a long time "S" was only used in the capital form; small "s" came later.
Here's what Carolingian miniscule looked like, note the r-like s: https://en.m.wikipedia.org/wiki/File:Carolineminuscel.jpg
And here's Old Roman cursive for comparison: https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Cu...
Small-s starts taking off around the 12th century or so, coexists with long-s for a while, before eventually replacing it.
That Carolingian script really has a lot of characters that can be easily confused, or relies on very subtle differences. b vs. h, i vs. j, i vs. l, r/s/t.