Okay, tryed on your demo site...
I can upload ( allthough i think it is not very logic to have the UPLOAD button first and then the choose uploadfolder, i would swap them but thats just me ).
The problem is with the tab choose existing, it just doesnt show any files.
i am using IE 8 btw
Oh, i tryed it on my own site and there i can upload new files also. Its the existing tab that gives the problem.
EDIT 2 >
Sorry UncleCheese, i was jumping conclusions...It just looks like upload is working, but when its finished with the upload the attached files remains empty an the file does not attach to my page in the CMS.
The file IS actually uploaded to the folder i choose but it doesnt show in the CMS.