Since Thin Slice Upload a Java Applet, your browser needs to have the java plug in installed. If the plug in is not available, the browser may prompt you to install it. Only trusted applets are allowed access to local files, so you should grant the requested permissions when the security dialog pops up.
Just drag and drop your files onto the applet and they will be uploaded. This demo is configured to accept a total file size of less than 100MB. The complete version can upload hundreds or even thousands of Giga Bytes.
The transfer can be resumed if it is interrupted.