The first document, SQRL explained, does a decent job describing how the system works. Having to download a PDF to read text on a website isn't the greatest of UX experiences.

The talking to seems to happen through a custom sqrl: URI-scheme. That's something that's supported by most platforms I know of; Steam uses the same mechanism to start installing a game you purchased in the browser, if I remember correctly.

Source code for several implementations is linked in the explainer document from the home page: https://www.grc.com/sqrl/SQRL_Explained.pdf

I'm not sure if the Windows implementation is open source, but the algorithm itself is.