The text-message preview on lockscreen of your iPhone is a value addition feature of iOS. This feature lets you see who sent what without having to unlock the screen.
While it's a useful feature for some, for others it was a visible breach of privacy because not many like personal messages intended for them to spring up (and show up) right on the lockscreen.
How to turn-off text message preview on iOS 7 lockscreen:
Step 1: Open Settings
Step 2: Tap on Notification Center
Step 3: Scroll down and Tap on Messages
Step 4: Switch “Show on Lock Screen” and “Show Preview“ to OFF
I don't know why a lot of people appear to be lost after upgrading to iOS 7 (may be they forgot the feature when they last turned it off on iOS 6).
The text-message preview feature is one of the few features that actually turned out be used by very few people. Although introduced to be a very useful feature, the lock-screen preview showed a lot of the message which could be a problem for privacy.
We really expected a minor change in the way messages showed up on the lockscreen. If only the name of the person who messages appeared (and not a part of the message along with), I guess many people would take that up.
A quick video tutorial that will walk you through how to turn-of message preview on lockscreen: