Start from the game address
Begin on the ORBIT website and follow Play to the game. Check that you have reached play.justonemoreorbit.com before adding an icon. The website explains the game; the game address is the place you want to launch directly. You do not need an ORBIT account or an App Store download.
If a friend sent you the link inside another app, open that address in Safari first. This makes the installation instructions easier to follow and gives you a consistent place to return to your game.
Add the icon and open it
In Safari, open Share and find Add to Home Screen. Apple notes that you can add the action through Edit Actions if it is missing. Leave Open as Web App enabled where that option appears, then confirm Add. Menu placement can vary with your Safari interface.
Launch the new ORBIT icon while you are still online. Check that the forest and game controls appear. Play a short run so you can also check sound, layout and the browser context in which your progress will live.
Check the two things installation does not prove
An icon is a convenient launch route, but it does not by itself verify that every offline file is ready. Prepare and test offline play separately before relying on it away from a connection.
Also check your discoveries and personal best in the newly opened app. ORBIT has no account-based transfer or cloud restore. Keep using the context containing the progress you care about, and do not delete its site data while investigating an installation question.