It seems that steam has registered a protocol with the browser which communicates with the local steam process. The following link might get you started with registering your own protocol in firefox at least:
https://support.steampowered.com/kb_article.php?ref=2087-MZES-9065
I would guess that there are similar links on the steam support site for other browsers.
The other part of this is probably going to be writing a simple local web server that can receive and respond to these requests. I'm not sure what language you are working in but an example for C# is the following: http://www.codeproject.com/Articles/36517/Communicating-from-the-Browser-to-a-Desktop-Applic. Best of luck!