When it comes to story writing the beginning questions that should be asked is why are you writing a story? What do intend the message for this story to be? Many of the more successful stories use morals as a guideline to develop a working world, it leads up to the important message. These stories constantly say something about the world you live in. This is shown evidently in animations as well books. All fables give you sense of perspective and teach an important lesson, this comes in as more important to a younger generation but it still speaks to older generations as well. To give a moral to a story also emotionally invests you to the characters, about their struggles to reach that lesson.
Two animations I have seen that speak of  this message is 'Sintel; with every revenge there consequences' and the 'Dam keeper'.

Sintel is a cgi animation that efficiently blends the fairytale element with reality and tells a significant story with a moral. The animation showed the struggles of a character and gave a lesson of how you can easily lose sight of your goals, that bitter emotions you harbour will inevitably ruin you and all the time will be wasted. Perhaps one of the more darker tales for the lesson learnt.

The Dam keeper was side project by the staff working on monsters university and the animation alone is inspiring to look at. The characters are simplistic and expressive based on painting and the message it tells is important. A young pig that is often bullied by his classmates keeps the darkness away but how you keep that negativity away when it surrounds is incredibly important.
The video that discusses the animation is linked above for in depth talk on how it was developed and the moral to it.