Before server side scripting (PHP, Ruby, node.js) there was server side programming.
One of the original interfaces between web servers and back-end processes was the Common Gateway Interface (CGI). It was introduced in the early 90s by the NCSA back-end team at the same time forms was introduced into HTML by Tim Berners-Lee (who was also at NCSA at the time). So forms was introduced at roughly the same time CGI was invented.
Initially a lot of people wrote CGI programs in C. I was one of those who had to do so as a homework assignment. Instead of a giant all-encompassing framework we wrote small C programs that read from stdin and print to stdout (we printed HTTP response, not just the HTML as per CGI spec). A website had lots of these small programs each doing one small thing and updated some database (sometimes that database was just a flat file).
Almost as soon as it was introduced people also started writing CGI scripts in Perl. So there was really no transition period between C programs and scripting languages. People simply stopped writing CGI scripts in C because it was faster to do so in scripting languages.