Other ligatures like tz and ch did not survive the jump into Roman letterforms, only ß for some reason.
By the way the now obsolete dual letterforms of s (ſ/s) was inherited from Greek. We’re lucky it was still in use in Leibniz’s time or calculus notation would be more confusing!
It is literally a ligature of the long s and z. This is unsurprisingly obvious in Fraktur, given the name, but you can see it in the Roman form ß: the upper concavity of ‚s‘ opens to the right while the upper concavity of ‚z‘ opens to the left (the bottom of Fraktur Z in both cases had/has a tail like g which is how you get the bottom part of the shape).
I still have a few books printed in Fraktur — each time I pick one up (which is not frequently!) it’s is very hard to read at first, but after a few pages goes back to being easy. Since this phenomenon takes a little “priming” it’s annoying to encounter a sentence or phrase of Fraktur on a shop sign or other short, random location.
Another commenter linked to an explanation of why a vowel shift caused “sz” to reflect “ss”. The character form and name are fossils (something I wish German retained more of — it’s one of the things I love about English).
The ß sticks around in German because it prevents some ambiguities. For example Busse and Buße are different words with different pronunciations.
Basically German orthography backed itself into a corner because of two rules: a vowel before a doubled consonant is short, and an s before a vowel is pronounced /z/. This leaves you with no clear way to write a long vowel followed by an /s/ sound followed by a vowel: VssV would make the first vowel short, and VsV would keep the first vowel long but the consonant would be pronounced as /z/. The ß solves this problem.
Some German. I believe it’s basically abandoned in Swiss German.
Just for completeness, there _is_ a capital ß: ẞ
This might help: https://thelanguagecloset.com/2022/11/05/the-story-of-eszett...
In short, it codifies a pronunciation that existed before a phonetic shift. The transliteration was codified after that same shift.