I backed up iphone 6 through iTunes (osx 10.9.5) and try to restore it on my new iphone13 but the iTunes can't connect to this new phone. The old iphone 6 does not charge.

As noted above you will need to be running El Capitan or better to connect to a recent version of iOS. If needs be see Locate backups of your iPhone, iPad, and iPod touch - Apple Support in case you need to copy the backup set to another computer. It may take swapping folder names with an existing backup to get iTunes to register the backup you'd want to restore.

It might be worth checking the docking port of the iPhone 6 for fluff. A little in just the wrong place can prevent charging.

tt2

Will the old iPhone work while connected to power? If so can you backup to iCloud?

For iOS 14 or later you need iTunes 12.8.2.3 or better running on OS X 10.11.6 or better. See How to upgrade to OS X El Capitan - Apple Support if required.

Assuming a suitable version of iTunes is in place then in Finder use the menu item Go > Go to Folder..., copy the following text, paste it into the dialog box and press enter:

/System/Library/PrivateFrameworks/MobileDevice.framework/Versions/Current/Resources

Unlock your device and connect it to USB. Double click on MobileDeviceUpdater in the folder that has opened. It should notify you of a software update. Install it and iTunes should hopefully recognize your device.

If these steps haven't worked you can also try reinstalling iTunes12.8.2.3.dmg (El Capitan/Sierra) or iTunes12.8.3.1 (High Sierra) while your device is connected and unlocked.

tt2

As noted in the earlier link the location of device backups on your Mac is ~/Library/Application Support/MobileSync/Backup/ where ~ is your users home folder and Library is a hidden folder you can reveal with Shift+Cmd+. (period). Once you've copied the contents of the backup folder over to a new computer you can see if the backup set is recognized automatically. If it isn't make a new backup of the iPhone 13, close iTunes or Music, and rename the folders so one replaces the other. E.g, If the new backup has the folder name <GUID> rename this to <GUID> New and then rename the original backup as <GUID>.

tt2

You're welcome. Best of luck. 🙂

tt2