I was excited to see how this worked, so I visited the page, clicked around to see what it did, disconnected from my network, and tried opening the page again in a new window. I got the usual browser error page: "You Are Not Connected to the Internet". So I connected to the network, loaded the page, then disconnected from my network, clicked the Demo button, and got: "Couldn't fetch demo SVG".
So by trial and error, I found that I need to load the page before disconnecting, and only use the 1st of the 5 buttons on the side (even "About" requires network connectivity). Then it works offline. Which is pretty cool!
But every native application I have works perfectly offline, and I don't need to do anything special ahead of time, or worry about which parts might not be available. There's a big difference between "some parts may work offline sometimes" and "entire app will definitely work offline always".
From decades of experience, when a "trivial" app struggles with demoing a feature, there's little chance it'll be widely supported among real apps. PWAs have been "any day now!" for 10 years now.
Sounds like the service worker didn’t register for you.
The logic that decides whether the browser accepts service workers is a little bit iffy. My vague recollection is that browsers default to something like “when you access the site for the second time, keep the service worker”. (Don’t quote me on that, and if anyone knows better, please correct me.) “Add this to your home screen” functionality will definitely make the service worker go. (That’s mostly for mobile browsers only at this time, though desktop browsers will eventually get it consistently available, as we’ve been promised for… hmm, about 8±2 years, I think.)
Privacy and blocking extensions may block service workers from being registered, too.
If the service worker is loaded, then it loads just fine with no internet connection, and the demo works too. The contribute and about links, being to GitHub, still don’t work.
I first actually discovered this completely by accident: I was offline, and thought “I need to shrink this SVG file, so I’ll open a tab to the SVGOMG URL and that error page will remind me to do it when I reconnect; huh, it loaded, must have a service worker. This is really great.” On later reflection, I realised that it being by Jake Archibald (a big mover in the service worker space) pretty much guaranteed it was going to work offline.
As a trivial example, https://jakearchibald.github.io/svgomg/ (which has no data storage requirements) works just fine offline.
Offline support is a banner feature of the PWA (progressive web apps) movement.