Solution No.1: Restart your iPhone
To do this, press and hold the sleep/wake button. Move the slider, and turn off your device. Again, press and hold the sleep/wake button to turn it back on. Visit the App Store and see if you can download or update the apps.
Solution No.2: Sign out from the App Store
This can help as well, and here is what you should do:
Go to Settings
Click on iTunes and App Store
Tap on your Apple ID, and sign out.
Go to the App Store again and sign in.
Solution No.3: Wi-Fi Settings
Try this: navigate to Settings, and click on Wi-Fi. Find your network, and click on the icon right next to it. Tap on Renew Lease. Go back to the App Store, and see if there is any difference.
Solution No.4: Update iOS
You can update the software wirelessly. Connect to the Wi-Fi network, and connect your iPhone 6 to a power source. Go to Settings, tap on General, and tap on Software Update.
You can also update your iPhone using iTunes. Here is the full guide that includes both methods:
How to update iOS on iPhone, iPad, and iPod touch
Solution No.5: Factory settings
As you probably already know, before you decide to do this, you need to back up your device. Then, follow these steps:
Go to Settings
Tap on General
Select Reset
Tap on Erase all Content and Settings
When the process of resetting is over, you should follow the steps provided, in order to set up your device as new.