4-dimensional objects can be described mathematically in detail. Equations speak for themselves to the mathematically trained. Actually visualizing 4 spatial dimensions is probably impossible for any human brain, yet you can develop intuition about 4 dimensions by considering an analogy with 3 dimensions vis-a-vis 2.
For instance, when a sphere “visits” Flatland by moving through its plane, the inhabitants first see a point, then a series of ever-enlarging circles, then a reverse of the process. The peak circle represents a cross-section at the sphere’s equator.
Analogously, a 4-dimensional hypersphere has 3-dimensional cross-sections. As it moves perpendicularly through our world, it appears as a point, then a series of ever-enlarging spheres, then a reverse of the process. A 4-dimensional hyper-human crossing through our space might appears as a series of blobs of flesh that seem to rapidly grow, shrink, move and jump around. This theme has been used in science fiction.
Note that here time is standing in for a spatial dimension. You can show the surface of any 4-dimensional object by considering a time series of 3-d cross-sections taken along its 4th axis.
Then there’s projection. Forget spheres and think cubes. An empty cube made of sticks can be projected to cast a shadow on a flat paper, which may be distorted in the process but still shows 6 faces, 12 edges, and 8 vertices no matter how you rotate it in the light. What you lose in projection, of course, is one dimension: 3 becomes 2.
Likewise a 4-dimensional hypercube (known as a tesseract) can be projected from 4 dimensions into 3 dimensions to show 8 cubes, 32 edges etc. (I had a cool animated gif of this but I can’t find it now. It looks like a small cube inside a large cube, connected at the corners by stretchy lines, constantly turning inside out as the the tesseract rotates in the 4th dimension.)
Similar to projection is folding & unfolding. A paper cube can be unfolded into 6 connected squares or folded back up into a compact cube. A tesseract can be unfolded into a cluster of 8 connected cubes. Salvador Dali rendered some of these in his paintings.
Another way of seeing 4-dimensional spatial structure arises in the mathematics of complex functions, where a complex variable z (which is is 2-dimensional) has a complex function value f(z), requiring a 4-dimensional graph to represent the function. Since you can’t actually make one, what’s commonly done instead is to make two separate 3-dimensional graphs (called the real and imaginary parts of the function) and somehow “imagine” that these are two different perpendicular projections of the same object.
Btw, cool links in your question!
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