What's so progressive about it?
Think of it as a web application, but with a native look and feel for the device it's being used on. The functionality is written once, and works the same no matter where it's being used, but on an Apple device it looks like an iOS app, on an Android device it looks like an Android app, and...I think you get it.
This gives an instant benefit that the user interaction is immediately intuitive. That in turn increases user engagement and therefore usage. People only use a handful of apps, so if you want your app to be among that handful it had better be easy to use as well as give them something useful.
The next thing to say about a PWA is that it can be installed on a user's device, as if it were a native app. It still uses a browser to run, but the address bar and other browser window features are hidden, so it looks like a native app. This gives your app instant access from the home screen on a phone or tablet, or from a desktop shortcut on a computer. The user doesn't even have to hunt for a browser shortcut. Handy.
Even better than just that, the fact that it's a webapp means that the user is always on the latest version. An installed app will be silently upgraded to the latest version without them having to worry about it. Progressive, indeed.
Another aspect making a PWA truly progressive is that it can work on any device, but it will use available features of devices as they become available. In addition, as the application itself is enhanced to offer more of its functionality offline, the app will feel more and more like a native app. Take an example of a ticketing website. As a progressive app, any purchased tickets can be saved on the device so they are available offline, so on reaching a venue with a dodgy internet connection the ticket can still be presented. Great user experience.
Security is a big issue, of course, and the application itself plus all data are guaranteed to be server over a secure channel, as one of the rules for PWAs is that they are served over HTTPS.
How can it benefit me?
There are several benefits, so its depends what you're after. Increased user engagement will help in lots of situations. Being able to claim guaranteed security (at least in the delivery; the app still needs to be written with a secure architecture) is reassuring to customers. Having an app that can be installed but is still really a webapp with just one code base makes the development more affordable in the first place. The option of continual enhancement and offering more and more functionality offline is a potential game-changer for when an application needs to be used in a place with patchy or no internet connection available. The possibilities are there to see...