If you want to learn more about this, search for PWA or Progressive Web App. That is the technology used here. When a website is adapted enough to actually work as an app, it will show the install thing; that is, the browser decides that, not the website, and it gives you some guarantees about how well the website works as an app (as in, was there any work put in to make it work well?).
RE: Has anyone used the 'Add to Home screen' feature of Android operating system for Peakd.com and Splinterlands?