Once you've installed the app, all you have to do is agree to a couple of permission requests if you're running Android 6.0 Marshmallow or above, then type your phone number. Once the code is there, you're inside and looking at your own sexy (or not) face through the front camera window.
If you fail to answer, a missed call notification will appear on your phone and let you either call back or message (SMS) the person back.
Compared to Facetime, Skype, Hangouts, and many other services, setup is faster and simpler, and usage afterward is more straightforward.
And yes phone numbers might be ripe for abuse and spam, but so are usernames and email addresses. I'd wager the reason is that Google wanted to do less, but do it better. Launch the app and you're greeted with your face from the front camera on your phone (we've already established it might not be sexy), a white bottom overlay with a button, and an overflow menu on the top right.
Part of what made Whats App so popular as a messaging client, and then again as a Vo IP alternative, was the low-entry barrier (plus free price and multiplatform support).
If you wanted to send an SMS to someone, you must have had their phone number, which meant you could also contact them via Whats App — and for free. Duo is using this same low-entry barrier for video calls.