How to restore a backup to iPhone 6s using out of date iTunes/OS X?
Are you able to borrow a friend or family member's computer to connect and restore the backup? If so, here's the instructions to get to your backup, and copy it elsewhere (it's ok to copy it to a Windows computer to restore as well):


In iTunes, go to the iTunes Preferences and then the Devices tab.
Right click on your backup and select "Reveal in Finder" (it may be that or "Show in Finder". I don't remember what version that text changed ;))
iTunes will open the ~/Library/Application Support/MobileSync/Backup folder. Each folder in here is an individual iOS backup named by the device UDID (making it a long folder name), and the Show in Finder should have highlighted your specific backup. Copy that folder to your USB thumb drive or whatever you're using to copy.


We'll need to copy that folder on to the other computer into the appropriate spot for iTunes to pick it up, so here are the instructions for Windows and Mac to get to that MobileSync > Backup folder.

Windows:


Windows 8 and 10: In the search bar type %appdata% (with the percents) and press return/enter. A file browser window will open to the user's Application Data folder.
Windows 7, Vista, XP: From the start menu, select Run. In the run box, type %appdata% (with the percents) and press return/enter. A file browser window will open to the user's Application Data folder.
From there, navigate to Apple Computer >  MobileSync > Backup.


Mac:


Open Finder, and select the "Go" menu at the top of the screen. Hold the Option key, and select the "Library" menu option from the Go menu. (On OSX Lion and above, Library will only appear with the Option key held.)
From Library, navigate to Application Support > MobileSync > Backup.


You'll want to copy the folder for your backup into the Backup folder, so the folder is sitting next to any other backups already on the system, very similar to how you found it on your mac.