Alright, I don't understand why the File Manager applet behaves the way it does, but my problem is solved. Here's what I found.
When I'm browsing the Internet on my home network with my laptop, I can connect to https://192.168.0.100:10000 and everything in Webmin works flawlessly, including the File Manager and SSH2 Login applets. However, if I log in to Webmin using https://fred.homelinux.org:10000, the File Manager applet does not open, but everything else in Webmin works perfectly.
Today, I took my laptop to work and connected using https://fred.homelinux.org:10000 and EVERYTHING worked, even File Manager.
Apparently, when connected to the same network as the server, the server IP needs to be used to log in to Webmin. Using the host name somehow causes File Manager to malfunction. It doesn't make sense.