I've been wondering how those work. I can understand how using a different DNS server can trick a streaming server which does the user distinction purely by DNS

I've been wondering how those work. I can understand how using a different DNS server can trick a streaming server which does the user distinction purely by DNS (in other words, only users from a certain region get the right IP address for the server). But I am unsure how the mechanism you describe works in detail (tricking the server to see a domestic IP, but having it send data to a foreign one).