Quote: "I presume these pages would then send the result back in a format uses by the C API system..."
I don't understand exactly where, but you seem to be misunderstanding the way it all layers together. Either that or I'm misunderstanding your terminology. ASCII art saves the day...
MySQL >sends results to> PHP
PHP >formats results into> HTML
HTML >is read by> MSIE
PHP, like a couple of other languages, sports an set of native interface functions for the MySQL DB. phpMyAdmin is basically a set of generic PHP pages which let you view/play with MySQL. The end result is a web interface to a MySQL server. There's nothing terribly magical about it, it's all just the glory that is webdev.