PWA simply put is a standardized way to add some context to websites and package them up so they behave as much like a native mobile application. A mobile application that you are used to install onto your phone or tablet most likely using an app store of some sort.
The aim of PWA is to provide a framework and tooling so that the website is able to provide features like push notifications, background updates, offline modes and so on.
Very neat. I’ve just today have enabled the PWA mode of this website, so you’re now free to add it to your home screen. But fear not: You won’t be pestered with push notifications or any background stuff taking place. It’s merely a more convenient optional shortcut.