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From Wikipedia on how to create a tesseract:

0 – A point is a hypercube of dimension zero.

1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.

2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.

3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.

4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

So far, this makes sense. However, the resulting image looks somewhat like this: enter image description here (Source)

Given the "Instructions" on how to create a tesseract/hypercube, we see that with each dimension added, the dimensions of the cube's properties increase by one. We can further notice that the dimension of the cube's volume is always one higher than the dimension of the face (like we have wih regular 3d-cubes with a 3-dimensional Volume, 2-Dimensional Face and 1-Dimensional Edge).

So if we were to add another dimension, we'd have 3-dimensional Faces. To this point, the image is correct. However, the volume which should be 4-dimensional is shown as 3D, too. I know we cannot properly draw 4D-space, but none of the tesseract illustrations I have ever seen even mentioned this.

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You are correct that there is a difficulty in any attempt to provide an illustration of a 4-dimensional object. The way this is usually handled is to use the idea of projection. The illustrations you have seen are projections into 3 dimensions of a wire-frame 4-dimensional object. And then that 3 dimensional projection was drawn by some suitable 2-dimensional convention, which is another projection.

To get the feel of this, consider what happens when you draw onto a flat sheet of paper a picture of an ordinary 3-dimensional cube. If you draw all the edges then you are drawing a two-dimensional picture which is the same as the shadow of a wire frame cube when it is illuminated with a beam of light and the shadow falls on your paper.

Now suppose we could arrange a light source off in some fourth dimension, shining onto our three-dimensional space. The shadow of a wire-frame tesseract would be three-dimensional, and if the light source is quite close to it then this 3-dimensional shadow would look like the image you showed in your question.

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