Oooh, I get it. It's like... pushing content to another system then? But the other system has it's own processing capabilities?
in short, upon 'casting' from a mobile device your device sends commands via wifi to the chromecast dongle to tell it what content to download and stream. The dongle then, on its own, connects to the necessary server and begins streaming the file. Your phone isn't involved in that process at all.
When you send a stop or pause or next command, for instance, it is sent via wifi to the dongle, which then executes said command on the stream it has been downloading. That's why you can continue using your phone or tablet for something else, or simply turn it off, while the content is streaming, because it only acts as a remote.