OK, thanks for that. The owners thing you can ignore. exFAT doesn't have the concept of owners of files, so that's not terribly helpful or important.
As for the "missing" stuff, were you expecting files to be there? I ask because you didn't actually say if you expected stuff to be there and it's missing, or you were just surprised that space was taken and no files listed.
If you thought it was empty and just want it back in service, use Disk Utility to format it again. DU can format exFAT just fine.
If you thought you had files and they are not there, you can try the First Aid process in DU to see if there is anything wrong with the structure of the drive. That should not harm the drive if the data is at all recoverable, but be aware that it COULD cause you to lose files if it determines that it thinks they are not recoverable.
A third alternative is to try to read the drive in a windows PC to see if it can see the files (if they are there) on the drive. It's possible that if you last wrote to the drive on a Windows system that it could be read on a Windows system. That's a long shot, IMHO, but probably worth trying.
And of course, there is the tried-and-true, eject and reinsert approach to see if it's just a glitch, but I suspect you have done that already, probably multiple times, right?
You can try showing hidden files on the drive. And another reason the space may be showing as used is if you deleted files on the drive but didn't empty the trash. You can check the trashcan to see if there are files there that need to be emptied to make the space available again.
I gave a link to an article on how to make hidden visible. The Trash can is also a hidden file that you can see using the trash can on the Dock, but only when the drive is actually mounted. Basically, there is a hidden file on the drive that the system opens when the drive mounts to show what's in the trash on that drive. If you unmount the drive without emptying the trash, that file stays there. But because the drive is unmounted, the Trash Can on the Dock no longer shows the files in the trash on that drive, just on drives that are attached.
I agree and I have found it invaluable for several years, and have also sent the developer an extra donation several times now when it does something outstanding. He doesn't exactly charge much in the beginning, and it is best to download it directly from the developer in order that he gets his fair share and it is not crippled like the Apple Store version.
A companion free utility EasyFind.app from Devon Technologies is also a must-have IMHO to complement FAF if and when needed.