This is probably the #1 misunderstood "fact" about fish handling. Too many people are taking this little tidbit too far and taking it to extremes, claiming every fish has a broken jaw and is killed if they're not held horizontally. That simply isn't true. Otherwise every single fish that has ever been caught would be dead, unless somebody has figured out a way to horizontally lip a fish out of the water.
The fact is, the larger fish, the greater the RISK of injuring their jaw when held at an angle. It is not a given, nor automatic that their jaw is instantly broken if they're held at an angle. But there's a potential. And if a fish is under 5lbs, it's not an issue. Simply put, there isn't enough weight on the smaller fish to put any stress on the jaw. You need leverage to put stress on that jaw. More weight = more leverage. And the greater the angle = the greater the leverage. It's simple physics.
While it's ok to hold them at an angle if they're under 5 lbs (and even larger fish - briefly), I put the limit at about 3 lbs, just be safe. Holding a small fish at a slight angle won't hurt them.
One other fact, holding them perfectly vertical will not cause any damage whatsoever, regardless of size. There's no leverage at play when you hold them that way. However there is a risk if they start wriggling and violently shake their body. The torque created increases the risk of damage. Again, the bigger the fish, the greater the risk. But again, it's not automatic. It's a potential.
It's irresponsible of the TPW to state a fish will "almost always be killed" by holding them vertically. It simply isn't true, and causing a lot of confusion as a result.
Here's a video with more information on it: http://www.bassresource.com/bass-fishing-videos/hold-fish.html