iPhone 5C running iOS 7 does not display Contacts
You likely have more than one contact "group".

In your case you will want to look two places.  First open the contacts app and note that using the "groups" button in the top left you will be able to 'show' a group that's not showing. That should get back the contacts that are "invisible" now.

Also navigate to 'Settings > Mail, Contacts, Calendars' and look at the "default account" in the Contacts section, which defines which "group" new contacts are stored to.