Your phone is connected to the Internet, but the software package to update your phone is not. It cannot connect to the update server at Apple....and that is why you see the message about "no Internet".
Make sure that you have a good Wi-Fi connection and the battery is fully charged. We assume that your phone has plenty of free space.
There are about a dozen different reasons why this might occur, and unfortunately there are no shortcuts. You have to work your way through the Apple troubleshooting document linked below to find the real culprit.
If your iPhone or iPad won't update - Apple Support
If this looks too intimidating to you, take the phone to an authorized Apple service shop to let them handle the update. You can find a list of authorized shops in your area using this link:
Find Locations