Just happened this morning. It appeared to be in guided access mode from the lock screen. There have been a few suggestions, a hard reboot worked for me. I see some new features in Messages so I wonder if there was an upgrade last night?
To do a hard reboot: Hold the top power button AND home button for 10-15 seconds (until the Apple Logo over grey screen appears). When it rebooted it asked me for my passcode, at this point I actually did a soft reboot (hold the power button to turn off the iphone) this likely wasn't necessary but I figured why not. Seems to be working now.
Other mentions, in the order I'd try.
1. Log in iCloud, Find my iphone, enter lost mode. (This didn't work for me saying it was offline.)
2. Drain the battery. (Stick it in a freezer? or go to Minnesota. I would hit the Apple store at this point.)
3. Force a reBoot with ReiBoot. (Not entirely sure I'd do this, check RDStokes answer.)
4. Restore from backup.
5. Restore iOS and set up as new. (Lawrence's answer.)
Hope this is helpful. The thought of losing all my info/data/passwords... uggh.