I've seen different causes:
1. Drive corrupted, or
2. Drive physically damaged, or
3. Non-drive components (SATA-to-USB converter) damaged
I actually ended up cracking open the USB case and connecting the drive directly to my computer for the exact problem you have now, though again no guarantees that's your problem so a last resort.
Before that you can try scanning with a tool from EASEUS to see if it can recognize files, though I ended up having to purchase the Pro version of the program (for a different defective drive that wasn't USB) as none of the free programs worked in my case.
There are several free programs you can Google, though if your FAT is damaged (but drive otherwise functional) you'd have to:
1) Have a drive to copy data TO, then
2) Scan and find all files (hours)
3) Copy files to spare drive
4) Format corrupted drive (FULL FORMAT)
5) Test drive
6) Create folder and copy files back again
Summary of Ideas:
1. Run diagnostics on drive (manufacturer tools)
2. Use HDD component only (last resort)
3. Rebuild partition
4. Recovery software to transfer files to another drive
*Note that if you try to REBUILD the partition and it's just the USB/Sata converter that's broken (main HDD fine) then you can create problems. Unfortunately it's a bit of a gamble.
**I'm also not certain what the DIAGNOSTICS prove. I suppose if it says the drive is okay then it's likely a corrupt File Allocation Table. If it gives ERRORS however it could either be a defective HDD or defective USB/Sata controller (HDD itself okay).
Sorry if this is confusing.