Here's a fun corollary to Kelly's answer:
The error can also appear if YOUR iOS DEVICE doesn't have enough free space.
Yes, the error's wording is completely wrong in this case. It says it's your computer that's lacking space, but, if you check
Settings >> General >> iPhone (or iPad) Storage and it's almost full, it could be your iOS device.
Just had it happen again... I followed the accepted answer, still no love, even tried installing iTunes as admin and running it as admin by right clicking and saying, well, run as admin, no dice. Tried on the D:\ drive, even though both C:\ and D:\ have over 150 gigs free and it's a 32 gig iPad. I get the "not enough free space on your computer" error complete with unhelpful "emptying the Recycle Bin" message.
Vaguely recalled this happening before, deleted a few gigs of stuff from the iPad, and now all is happy. I just completed an encrypted local backup.
Again, the error said my computer didn't have enough hard drive space & said to empty the recycle bin, etc. But it was, indeed, the iOS device that needed free space for the backup to execute.