For those who say they don't get the story, or the story is wierd, I think the story is about chasing your dreams. Sounds really simple, right? Here's a breakdown of how this is the case:
The kid starts off as a child, leaves his house, is careless and directionless. This is shown by him playing hopscotch and messing around. He wanders into the woods, and this is where he sees the white rabbit. For the first time, his eyes open wide. When he tries to catch the rabbit, and hold it, it turns and runs away.
The Rabbit is drawn in a stylized way that makes it look ghostly, heavenly, or symbolic. This is meant to show its not a literal white rabbit. This is even more obvious further on.
The boy chases the rabbit and loses its trail when he comes to possibly high school possibly middle school, and looks around. A girl pushes him towards a rollarcoaster and he sees flashes of the white rabbit in her, giving a clue that he might be able to find it if he rides this rollar coaster, but the rabbit is turning away, so seemingly he is not getting closer to it.
He rides a rollar coaster meant to represent the crazy emotional years of school until he reaches a high point right before the drop. Before dropping with the other kids, he jumps out of the rollarcoaster and tries to touch the light above him, which is the same silver color as the rabbit. However, he falls down through swirling darkness. This can be interpreted any number of ways, but imagine the light as his goals and dreams, and falling down into darkness as depression, losing his way, being confused, or failing.
Then a significant thing happens; he gets caught on strings, almost like he would be made into a puppet. But he immediately shakes free of these strings, with a look on his face like he is bothered by them or even disgusted. He refuses to be controlled at this formative time in his life, and therefore falls into a boat on a sea with no sight of land. This is again symbolic, and we can see how he feels about this by his expression. He looks disappointed, frusterated, but still determined. He remembers the rabbit and that he wants to follow it.
As soon as he gets to land he sees the rabbit ahead of him, and immediately starts chasing it. Notably, instead of running after it with the carefree skip of a child as he once did, he is sprinting purposely with the strength and purpose of a young adult. He is determined to reach the rabbit and not lose it again. Cities spring up all around him, so different from the peaceful forest, and once again, this can be taken literally or symbolicly, as either his setting or residence in the city possibly. Then a pile of paperwork, symbolic of a job with meaninless and unfulfilling or endless work, falls upon him.
This is like the string moment. He stares at the papers for a moment, which are heavy and slowing him down, making it so he can't see or catch up to the rabbit. A moment later he decides and tosses the paper away, choosing to give up the security of a job for chasing his dreams, or even choosing to ignore the work descending upon him to do work that he'd rather do or to pursue his goal. This can interpreted several ways but its obvious when he throws the papers away he is elated by this and able to run faster.
After this, the rabbit leads him into a forest, very much like the one where he first found it, the place he started as a child. It stops on its own, and lets him approach it and pick it up, acknowledging him. In other words, through all these trials, he managed to keep chasing after the white rabbit, his dreams, his goals, his aspirations, and finally, he mainaged to catch it. In this moment he looks through time back to his childhood self, and the child he used to be smiles and nods to him, acknowledging that he never gave up on his dreams from so long ago, and finally managed to achieve them.
The title of the movie "Don't Look Back' isn't necessarily about not looking back at all, because the character does that, but rather about having to keep moving forward with your life through everything that happens and continuing to chase the rabbit until you catch it, rather than stopping or dwelling on past failures. This is a symbolic story of success and all the hardship you have to go through in order to achieve your dreams, but the way that it is very possible to pull through if you don't let the strings hold you and make you a puppet, if you don't let the rollarcoaster take you along with the rest of the crowd, if you dont' get lost on the sea, if you don't give in to the pile of paperwork that bogs you down, and if you dont' simply lose sight of the rabbit altogether.