Can't install apps from App Store on iPhone with iOS 7
I would suggest trying the following options:


Press and hold the power button and the home button until the Apple logo appears. Once it restarts, try downloading.
Reset your router from which you are accessing your Wi-Fi.
If the issue persists, trying downloading apps from a different Wi-Fi network or cellular data.
If these fail, Reset Network Settings (Settings > General > Reset) and do steps 2 and 3.
If the issue still persists, Reset All Settings (Settings > General > Reset) and try again.
As a last resort, take your phone to an Apple Store.