"The Appalachian Trail isn't very dangerous"

Personally, I think this is largely true. Predatory animals, extreme weather changes at elevation, and very remote hiking locations aren't as common on the AT compared to a lot of the trails you'll find out west, for example. 

But you're still out in the woods when you are on the AT, and failing to respect the dangers of any outdoor activity and going outside of your ability and experience level (being unsafe) can and will get you hurt or killed.

There are places throughout the trail, namely the larger wilderness areas like the 100 mile wilderness, where you could be in some serious trouble if you screw up. Most of the Appalachian trail, however, is not 'far' from civilization.

That said, hiking and backpacking are inherently dangerous activities in that a few hours hike is pretty far from anywhere with an injury or when exposed to harsh environmental conditions. Many people that either get hurt or die after being lost are found within a few miles of a trail head or where they parked their car to hit the trail. 
You can still manage to go "stupid light", failing to maintain an acceptable level of safety through your equipment load-out, in the most favorable of hiking conditions. Note for example that some people think, and tell others, that you don't need rain gear on the Appalachian Trail in the summer in spite of the fact that it is possible to become hypothermic in 80 degree weather with persistent rain. 

In regards to animal threats: black bear, wild boar, moose, snakes, and ticks and other insects are all worthy of respect. Always research the area you are planning to hike in to see what hazards, animal or otherwise may present themselves on your trip.

With experience and skill nature seems a lot less scary for us hikers and backpackers, but we shouldn't ever forget the brutal smackdown nature can dish out. Having the knowledge, skill, and equipment to stay safe when things go wrong is always important, even on the least dangerous of hikes.