Either the hard drive inside the case died or the USB controller in the case died.
If just the USB controller died, and you do not mind voiding your warranty, you can open up the case, remove the hard drive (it should be a standard SATA drive), and connect it inside your computer to copy the files from it.
If the hard drive itself died your files are gone. But you are now wiser because you know to NEVER have just one copy of important files. An external hard drive is handy to store and move files, but you should have copies on a PC also.