Tap each part on the left as you collect it. When every part in a row is checked, that recipe's result glows green — you're ready to fuse it in-game.
Tap the circle next to the result once you've actually fused it. The whole row strikes through and fades, so your list only shows what's still left to do.
Your checkmarks save automatically — right in this browser, on this device, no account needed. Close the tab, close the app, restart your phone: they'll be exactly how you left them next time you open this page.
To keep them saved for good: open this page in Safari (not a private/incognito tab) and use Add to Home Screen so it behaves like its own app. Your progress sticks around unless you manually clear Safari's website data for this site, or switch to a different browser/device — checkmarks don't sync between browsers.
To start over, use the Reset All button at the bottom of the page. That's the only thing that clears your checkmarks — closing the page never does.