There is nothing to download: an icon lands on your home screen that opens the site with one tap — no browser address bar, full screen. The same site, with the same account, just closer to hand.
The other reason is notifications: on an iPhone and an iPad the browser can only send them from an app put on the home screen.
The icon then sits among your other apps, and starts from there.
If Chrome offers the install by itself in a bar at the bottom, accepting that is enough.
In Chrome and Edge a small install icon appears at the right end of the address bar: click it and the site starts in its own window, with its own icon. On a Mac, Safari does the same with the File → Add to Dock menu item.
On a phone, scroll to the bottom of the page: at the top of the footer, under the logo, you will find the "Install the app" button. It brings the install offer back even if you once dismissed it.
This button is only visible on a phone — on a computer look in the browser's address bar — and it disappears as soon as the app is on your device.
