I think of it this way: if you try to draw a 3D cube on a 2D screen, what you really have is a 2D image. If oriented correctly, and nobody told you anything about that cube, you might mistake it for a 2D hexagon. If instead that cube were rotating, you would see the change in the shape of the hexagon and realize that the image is supposed to be a cube.

But the reality is, it's still a 2D image. It's just a representation, and context and animation help you get back to the notion that it's a 3D image.

Now, think for a minute about how the screen works. There are rows and rows of pixels. Each of those rows is actually a 1-dimensional image, a slice of the whole. If you were to only see one row of pixels at a time, you'd not be able to figure out what the image is. Maybe if you knew certain parameters about the image, that it was of very low fidelity, simplistic shapes, you'd be able to analyze one row after another and figure it out. You'd have extreme difficulty just looking at a 1-dimensional image, or a serious of such, and figuring out it would be of a 3-dimensional object. To really see the image, you again need it animated. The rows are scanned out very quickly, and your brain blurs it all together.

So, when you're dropping from N to N-1 dimensions, you lose something, something that you can sort of get back if you add some motion in. When you drop from N to N-2 or more, you lose so much that you have a really hard time figuring it out.

Now we can finally come back to the tesseract. It's a 4D object. We're trying to display it on a 2D display. It's not going to work out too well. You've really got to stare at it, think about what is going on, understand that you're dealing only with a very low fidelity, simplistic shape. Analyze it, rotation after rotation, and you might be able to figure it out.

Or, in other words, it would be a lot easier if you had a real 3D display.

I had a math professor who put it this way: Just think of a general n-dimensional Euclidian space, and then set n=4.

It's tricky because there is just no way to show 4D phenomenon without a lot of sacrifice. I ended up using a combination of "perspective" for size and differing transparency and a fair amount of movement. All of these are artificial of course.

The explanation on the page takes another approach: all black and white line drawings and a lot of brevity.

But it's really a hell of a lot of fun tying to figure out how to present impossible to visualize phenomenon.

And you are not remotely too dumb: if my story didn't come through it's my fault.

You're probably trying to think about it in spatial terms. But it's impossible to understand a 4D object with 3D spatial reasoning. I would say ALL humans are too "dumb" to grasp this concept. Of course the math is fairly simple, but there's a difference between understanding the math and truly understanding what a world with 4 spatial dimensions would be like.

I struggle with picturing 4D rotating objects, I find it easier to deal with slicing the object up and picturing that. To clarify if you think of a single picture from an MRI it's a slice of a 3D object and you can move up and down the object to see each slice. If you move through all the slices you can get an idea of the 3D shape. With the 4th dimension it's similar except each of the slices is a 3D shape. So a 4D “sphere” sliced up this way would look like a 3d sphere that gradually got bigger until halfway through the 4D “sphere” then gets smaller again. You can also somewhat consider it a 3D object that changes shape with time but since time is a dimension you can go back and forward along it. That's how I picture 4D objects which is admittedly not as well as some people do.