It works same as most other debugging tools, and same as your local debugger.
Locally, the debugger is a separate process that connects to the standard java debugger. It is listening on a local port e.g. 8090 by default. Anyone can connect to this port and debug if they know the corresponding debug protocol. The desktop modeler of course knows this protocol so it is able to connect to this separate process via port 8090 and can show you some of the results graphically in the modeler.
Remote debugging works exactly the same, the only diference being that the debugging process is running on a remote machine, so the access port is not localhost:8090 but something else e.g. https://example.mxapps.io/debugger
Hope this helps,
PS: I am fairly certain that there is check to make sure that the version of the model that is running in the cloud and the one in your local modeler are the same, to prevent strange behaviour.
PSS: the password is there for obious security reasons.
Thanks a lot for the quick response, Andrej.