The biggest usage is probably not the tool itself, but the library.
It's the de-facto-standard-HTTP-library (although it does more than just HTTP). E.g. if you have a PHP script that downloads some data from another webpage it very often does this with curl.
It's used to send an http/HTTPS reqeuests, controlling every aspect of it, form headers to cookies, to ignoring/not SSL certs, and has advanced debugging option to show you the entire dialog (-sv) and use a different IP address (--resolve to test you firewalls, LBs etc...).
"curl | sh" is no worse than "wget && tar xf && ./configure". I have yet to see anyone who knows how to audit a configure script generated by GNU autoconf, which is generally a multiple-tens-of-thousands-of-line monstrosity that generates a bunch of C files and compiles and runs them. FUD about "curl | sh" isn't rooted in any sensible security modeling.
As a general rule of thumb, if you think everyone around you is independently doing something stupid, you should first pursue the hypothesis that it is your reasoning that is flawed and not the entire rest of the world's.
Almost all of the examples on that page use curl https:// | sh. Which, again, makes it a superior option to wget && gpg --verify && ./configure; I have yet to see anyone who is better at PGP fingerprint verification than their OS's SSL stack is at TLS certificate verification. (There are a very small number of people who are as good, but not better.)
It's the de-facto-standard-HTTP-library (although it does more than just HTTP). E.g. if you have a PHP script that downloads some data from another webpage it very often does this with curl.