We have to restore your iPhone from iTunes on your computer. This will install a fresh software on your iPhone. Follow the steps below;
First download and install latest iTunes on your computer from here : https://www.apple.com/itunes/download/
Then on the iPhone, go to Settings > iCloud and turn off Find My iPhone
Then connect the iPhone to your computer with the USB Cable.
Open iTunes application on your computer.
iTunes will recognize your iPhone. Select your iPhone in the top left (select iPhone icon)
Then in Summary section, click Backup and make sure "To this computer" is selected. Make sure the backup is done properly
Then click Restore button and let it restore. It will take some time to restore the iPhone .
After restoring, you will get option to restore your backup. Select Restore from iTunes backup and all your files will be restored. Now try to download the apps.
(This will take some time to complete. So start this process only if you are free for a while. Doing this will replace old corrupted iOS with new one.)