How does the communication from the computer to the phone work?

No idea. Presumably a push notification. Within less than a second providing your email (on fast US data) you get a prompt on your phone that shows the browser model and approx location and asks you to approve or disapprove the login.

They initially rolled it out as a 2FA option, then as an optional for 1FA.

OK so you have to actually type your email into the computer.

Exactly. Google already asked for email alone and then password on the next page so that they could support sending enterprise emails to custom login systems.