The maze is a hypercube; in the same sense that a diagram might represent a three dimensional structure by showing some two dimensional slices one above the other, this maze represents a four dimensional structure by a grid of two dimensional slices.
In eighth grade, when I first wrote this, I was very interested in higher dimensions, and I tried to learn to visualize in four dimensions; I never really succeeded, but I did learn to visualize the shifting three dimensional cross sections of hypercubes passing through three-space at a couple of different angles. Visualizing higher-dimensional objects is hard (impossible, by some sources), but doing math or programming in higher dimensions is not by nature more difficult than doing things in two dimemsions.
Does the maze look flat? Is it really four-dimensional? Let me ask you a question. What can you see that doesn't look flat? All the images you see are two-dimensional. A movie or a 3D game doesn't look flat because it represents a three-dimensional area in a two-dimensional way, and you learned as a child to perceive that as three dimensional. This maze also represents a four dimensional maze, just like a flat schematic diagram represents something three-dimensional. The picture on your screen is two dimensional, just like every other picture on the web, but the maze that's represented is four-dimensional.