Rob's dead on, but this is a slightly different treatment.
Imagine you are standing in a field. Out in the field is a post. You know this because you can see it.
If you walk to the left, from your original position, and face the post again, you can still see it.
If you walk to the right, from your original position, and face the post again, again it is visible.
The obvious implication is that there was a ray of light going in one direction that hit you in the center, a ray of light that goes in another direction that you saw from the left, and another ray, from the right. By carefully choosing which rays (ie pixels) from each camera, you can reconstruct novel images using whichever rays you like. You know which rays you are choosing by the distance between the points of capture. The "light field" is another way of saying "a big bundle of rays with an X-Y position and a direction".
Matti and I did this in the crudest and fastest way we could, just to get up to speed. A lot of other people have it covered better, and some people have built much smaller, more sophisticated cameras
. I particularly recommend the tutorial materials from Todor Georgiev