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.
But I clicked around for a while and only found some videos that might show me what SQRL does, but I didn't actually feel like watching a video, so I still don't know.
Based on what I read, it sounds like snake oil so far. Too good to be true. I don't remember the phrase, but it suggested it would be my last password solution ever, which just sounds like snakeoil.
Also, it seems like only the Windows implementation is mature. All other implementations are by third parties and are marked as not being complete.
EDIT: A user posted a very nice write up of it in another comment: https://news.ycombinator.com/item?id=26314472
Right.
> A major downside is difficulty of backup. The private keys are locked inside the hardware and cannot be accessed in any way.
That's a feature, not a bug. You buy at least 2 keys (1 backup), ideally 3 (2 backup).
As for SQRL, I never took anything serious at grc.com/Steve Gibson. He was all about snake oil 20 years ago, and probably still is.
Why is this better than WebAuthn? It looks almost the same but WebAuthn has much more support. It can use software-defined keys like Krypton though certainly it would be good for browsers to have standard APIs for this stuff.
https://sqrl.grc.com/pages/introductory_questions_and_answer...
It says "Using a SQRL app, a master identity is created and shared among the devices. Websites which support SQRL logon trigger the app to securely identify the user."
How does this happen? How can a website talk to an app?
Also where is the source code?