4D Conceptual Work
Move a cube perpendicularly to itself, and a tesseract will emerge from it
The cube is representing space as we know it. A direction perpendicular to it is not familiar to us, and we have accepted this restriction until mathematician Bernhard Riemann presented a new view on geometry in 1854. He generalised Euclidian to NonEuclidian geometry, enabling these reflections.
To experience hyperspace, we refer to analogy. Single elements of the dimensions [0D] up to [3D],
points  lines  planes  spaces,
are transported through line of evidence into hyperspace. So, a hyper cube called tesseract can be obtained from a cube. This way, we may enter hyperspace by means of imagination at least mentally.
A teseract is a 4dimensional cube, or, to be more precisely, a hyper cube.
It has 16 angles, 32 edges, 24 squares, and is limited by eight 3dimensional cubes.
This depiction is a parallel projection from hyperspace [4D] to space [3D] and projected from there on a plane [2D]. According to the chosen perspective, other perceptions are possible as well.
Training:
For the following, the meaning of "Folding"and "Unfolding" has to be introduced.
Unfolding of a cube
Unfolding a 3D cube and dismantling it into its parts results in a graticule consisting of 6 planes [2D].
This depicture is showing one out of 11 possible forms of graticules.
Such a graticule may serve as instruction for construction of a cube as well.
Folding of a cube
Here 5 out of 6 planes [2D] have to be folded (bent at their lines of contact) and thus to be moved through space [3D]!
Unfolding of a tesseract
If your visitor from 4th dimension is busy just now, let him do this later.
To do the task ourselves, we have to do some training for the time being.
Tesseract from another perspective
 In this view, we perceive a small cube inside a large one. Training:

Rotation of a Tesseract (a 2D projection of a 3D projection)
 The eye is marvelled, and the mind is wondering

giving it more "flesh on the abstract bones"
This is only one of the possible cycles.
Very often, the others are coming inbetween.
This is one of the phenomena of curved space, for this is what we are dealing with in 4th dimension.
The comprehensible results:
During time lapse of a tesseract's rotation
and vice versa....
There is a lot to think.
Mathworld.wolfram.com links to explore:
2D  3D  4D  general 
Spirtual View
From this conceptual view of hyperspace, the 4th dimension seems to have some qualities to make a higher reality (one above our 3D) a little more understandable, regarding
see chapter Dimensional Views [ 01.01 ] Eye In The Hand
and more to comeIt seems
 we cannot expect to enter this 4th dimension with our bodies  without a permanent change in orientation.
 our mind is not able to understand a perpendicular move to space (3D) intuitively.
 higher spiritual qualities may belong to the nature of 4D.
Zero Dimension [0D]
Zero dimension represents a state of zero degrees of freedom in movements.
It seems to be optimal for
 meditation (zero movements of body & mind) and total relaxation (kayotsarg).
Zero and 4th dimension have high spiritual qualities.