Does anyone know how I can transfer files from my local computer (Windows XP) to a Windows 2000 server, using Remote Desktop Connection?
I tried going into Options>>Local Resources>>Local devices and resources>>More...>>Clicked the checkbox next to Drives, but my local drives don't show up on the remote server under My Computer.

Can anyone advise on how I might make it work?

Actually what u have done must work, But i cannot exactly tell what is that had happened in ur case. So try using the filezilla server on 2003 machine and using client upload the files on to the server.