You run the backend server, it’s entirely up to you to decide how and where you want to run it, Twitch has nothing to do with that. Because extensions can be used at any time you will likely want your server to run your backend code 24/7.
To communicate with your backend from your frontend you could either use a HTTPS request, or a Websocket connection. In the Hello World example, the frontend files use jQuery to make an AJAX HTTP request to the backend service, in a live extension you could do something similar, except you would point the URL to wherever you are hosting your backend server.