Mobile Safari has "reader view" build in for most sites, but it doesn't work on every site. It works on most blogs and news sites though. You can find the button left from the URL.
Desktop Safari processes the page to try and identify if there is a block of text that might look like prose. If it finds it, you get the button. If it can't, you don't.
I assume mobile Safari does something similar.
Yep, it's an algorithm site authors have no control over. It's definitely useful, but not something to rely on as a dev.