How does my phone start installing apps so quickly when initiated from the play on-line store? When I go to the Google Play on-line store and install an app, how does my phone know to install it? Is it using push notifications? Quite often it happens within a few seconds so presumably it can't be using polling.

I just don't understand the nature of the connection between my phone and Google's servers.
Yes, it's using a push notification via Google Cloud Messaging (GCM), the same as Google+ Hangouts, Google+, Gmail, and some third-party apps.