Mathematically speaking, to say that a space has n dimensions roughly means that you need n coordinates to refer to a position on that space. In two dimensions (like your computer screen) a pixel is uniquely determined by its x and y coordinates; in three dimensions, a point also has a depth, and so you need x, y, and z coordinates to determine its position (hence a z-buffer). So to say that this game takes place in four-dimensional space just means that the world is a space that requires four coordinates to describe, say x,y,z, and w. Since the "visible" part of the world appears three-dimensional (albeit projected on a 2-D screen), what is presumably happening is that at any given time you're seeing a "slice" of the 4-D world -- probably you start off seeing all the objects contained in the slice where w=0, and you can "shift" objects into other slices, say where w=1 or w=-1. In the example of connecting rings, the player is moving the one ring from the w=0 slice into the w=1 slice, which looks to be pretty empty; then he can move it around as he likes before taking it back into the w=0 slice, where it's now locked into the other ring.
I would simply think of the game world as a bunch of parallel 3-D worlds. Exactly like a Linux (or Mac) desktop with 4 workspaces is really four parallel 2-D worlds, which have little to do with each other, except you can move windows from one to the other.
Non-mathematicians generally get hung up on the need to assign a physical interpretation to the fourth coordinate. But if you think of a four-dimensional space simply as the set of all four-tuples of numbers, then you've divorced the space from its interpretation as spacetime or whatever, and it becomes much easier to think about.
