Check on the top left section of the phone on the side. It's above the volume keys. You'll see this little bar that contains your nano sim. Do not pull the tab out or push it inward to eject the sim. Instead, flip that little bar like a light switch. It switches from silent to ring mode on the phone. Mine gets caught on stuff all the time and puts it in silent mode all the time.
That is not the layout of the iPhone 6 (SIM card tray is on the right side of the phone where the lock/power key is. The silence switch is on the left side above the volume keys).
If you have made sure the mute switch hasn't been turned on accidentally make sure "do not disturb" isn't on in settings. Also try resetting your device (hold the home button and power/lock button simultaneously until the Apple logo appears on the screen). If that doesn't work try disabling notifications for messages in settings, reset the phone as mentioned above, and then turn notifications back on in settings. These are the tips that came up in several articles I saw when I googled the issue.