Okay I see the confusion.
So socket.io comes in two parts. The server side and the client side.
Server side, its just a simple library that you control on your server. I'm going to use node.js as the example for the backend frame work... you can use whatever backend frame work you're comfortable with like PHP or Django (python) etc. So you have your server, with socket.io, it will handle all the routing and stuff for you once you initialize it in your server code. You handle broadcasting and receiving messages from here.
The client side that you're seeing on the website is just regular javascript code you see on most websites that simply creates a connection to your server. All that they're doing there is providing you with a CDN which is just a link to their javascript so you don't have to waste your own hosting resources to serve it to the end user. They also give you another link so you can just serve it yourself instead of using their convenient one. Again, this is just for the front end... with hyperPad, you don't need to worry about this side at all. All the client side code is presented as behaviours.
With the client side you handle receiving messages from the server, and emitting messages to the server. It just provides a realtime link between the end user and your server.
Edit:
Finally, when they are saying Socket.io servers... they're talking about your socket.io server. They're just saying, "Your socket.io server provides this handy convenience if you don't want to deal with our CDN"
