I was curious how this works. Does anyone know how?

If I get your question right, in brief, the hash part of a link (for example, 1f39cabf0794ee984490bee454acbf60 in https://www.classicube.net/server/play/1f39cabf0794ee984490bee454acbf60/) is the result of md5(server_ip + ":" + server_port) function. By loading this page, ClassiCube site provides all neccessary info (server IP address, server port, your username and mppass (based on your browser's cookie files)) in order for the web client to start.