I restored my phone from a backup and the apps were synced with iTunes, but the App Store says it's trying to download apps that are already installed. It won't go passed the loading wheel and it's doing it with multiple apps.
Usually such App Store issues are not easily fixable.
Something you may try at first is the following:
Restart your iDevice
Sign out and back in with your AppleID I'm the App Store
Sync your iDevice using iTunes
If all the above possible solutions didn't work, you may want to restore your iDevice, so first of all you'll have to:
Make a Backup of your iDevice and save it either on iCloud or on to your Mac/PC.
Erase all content and data of your iDevice through Settings > General > Reset > Erase All Content And Settings.
Once you erased your content, and you initialize back your iDevice, when asking wether you want to setup your iDevice as new or restore a Backup, choose the second option, and select your backup.
For a reference and an official guide about successfully performing a restore, you may want to take a look at this Apple article.