It is open source. Your browser retrives all the JS and html sources. Even if it were somehow closed source, it doesn't really matter. You can view web requests and other, and see for yourself.
Theoretically, you should be able to block any outgoing connections to the website domain (after it loads all necessary HTML and JS), and it should still work fine. Saying "theoretically" because it might attempt delay-loading some static assets, but you get the idea. No idea how this specific website functions, as I haven't looked deep into it yet, but that's the principle behind "static websites" like this one.
I can view web requests but let's say on the 365th web request it sends my API key to an unknown location. How would I know if this was closed source?
Your example is valid, but it's like asking "how can I know that my iPhone doesn't record me while I'm sleeping?"