How do I restore iPhone 6s stuck on connect to iTunes screen/Apple logo
The network error means iTunes can't get the files it needs from Apple so there's nothing to do on the phone at this point.

The fastest way to sort any iOS device that says “connect to iTunes” and your preferred iTunes isn’t working is to take your iOS device to iTunes on a computer that you know works. Apple stores maintain multiple installations of iTunes ready to restore devices. No data sync happens when you are at the “restore screen” but you can check for local backups in iTunes when you’re done if you like. 

As to fixing your specific error, can you take your PC/Mac to a different network or take the phone to a different computer? Sometimes that's far easier than diagnosing one iTunes that isn't working for a rare event like a DFU restore.

If you want to troubleshoot your iTunes, I would start with:


https://support.apple.com/en-us/HT204770


It's quite detailed, so you will want to make note of the exact error message and be sure to follow steps like updating the computer and iTunes as well as making sure anti virus and other security software and network blocks aren't the cause of network errors.