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.