The problem is because you have restored your iPhone from a previous backup that contained old app data. The old app data was restored and it contains data from a previous version of that app. When you upgraded to the new app, the old app data didn't upgrade correctly so you see this warning to connect to iTunes.
The fix would be to try a restore to factory settings and then a restore from backup again. If the problem persists even after restoring to the same back up then your Backup may be corrupted. You can try to restore to the most current back up or you can restore as a new phone.
Please note if you purchased a new iPhone and never had an iPhone previously then you will need to sync your device to iTunes > Back up your iPhone > Restore your phone.
Here is a link discussing this issue: https://discussions.apple.com/message/16463435#16463435
If you are still having problems, please submit a ticket to our support team.