Why would all places appear to be at the center of the universe's expansion? I just watched an iMax planetarium video that claimed that no matter where you are in the universe, it will appear that you are at the center of the universe and all things are expanding away from you. 

What the heck?? Can someone please explain this?
This is a famous analogy, which was prbably given to you in the video. Imagne that you are on a ball of dough full of raisins. You obviously perceive yourself as the center. Then, the dough starts to grow. The raisins are still the same size, but there is constantly more and more dough between them. From your position it would seem like there is always more and more space between you and everything else. You don't see yourself moving, but you see everything else moving away from you. This effect would happen anywhere in the universe, not only at the center. In fact, you would think - correctly, relatively speaking - that the center (if you can see it) is being pushed away from you. So basically, it all comes down to your point of view.

This expansion is thoroughly studied presently. The most common theory is that the universe contains a giant amount of energy (called dark energy) that is so far undetectable and that is causing the the space to expand. Think of it as yeast. There are other theories that are much more complicated, which a non-professional such as I wouldn't be able to explain. (For instance, there is one that suggests that our universe adheres to other universes, like soap bubbles, which adds space). But dark energy is the most likely candidate, so far.