It opens like an app, without the browser bar, and keeps you signed in. There is nothing to download and no app store involved.
In Chrome or Edge, click the install icon at the right of the address bar, or open the browser menu and choose Install.
Safari on a Mac can keep it in the Dock: File, then Add to Dock.
Want it on your phone instead? Scan this with the phone's camera and these instructions open there.
Installing adds an icon and nothing else. It stores no mail on the device beyond what the browser already keeps, and signing out of the app signs out everywhere it is open.