4d space, a.k.a. hyperspace, is space where four coordinates denote a point. Hyperspace is not fundamentally more complex than ordinary 3d space, it's just very hard to imagine visually. For now, just don't try that. Instead, use what you know about 3d space and find the 4d analogies.
|3d||(x, y, z)||space||cube||up||down|
|4d||(x, y, z, w)||hyperspace||hypercube||ana||kata|
Particularly helpful is trying to explain the concept of 3d space to imaginary 2d beings, like in Flatland.
One way to do that would be to present the 2d beings with a computer-generated projection of a 3d scene, and enable them to interact with it. Imagine a 2d being living in the 2d world of your computer screen, playing this 3d game. Its experience would be very simular to yours, when you attempt the 4d version.
A computer model of a 4d scene can be projected into a 3d-stereoscopic image. If the 3d image is transparent, and you as a 3d spectator are in it, then you are as close as you can get to enjoying it as what it really is: a 4d scene!
Each time you encounter a problem with the 4d hypercube game, you should go back to the 3d version and figure out how its flat 2d projection tells you what to do. Then apply the same rule to the 3d-stereoscopic projection of the 4d hypercube. Most of these rules are very basic. You know them, but never give them a second thought.
Here's an example. If in the flat 2d projection of the 3d cube the cursor is inside the 2d window (a square on the cube), you know that in reality the cursor is either in front or behind it. To pass the cursor through the window, you have to move it forward or backwards. Likewise, when in the 3d-stereoscopic projection of the 4d hypercube the cursor is inside the 3d window (a cube on the hypercube), you can pass through it by moving the cursor forward or backwards.