So, people have trouble envisioning spaces.
The average person has trouble viewing things in three dimensions; sure, you can tell distance- but can you envision objects as existing from all angles? Can you, whilst paying attention to everything else, be aware of the fact that they exist as more than just the flat image you see?
People see stereoscopically; that is to say, that while things appear to be three dimensional, it's actually a trick caused by the combination of two side-by-side 2D images, one from each eye. This is most easily noticed by looking past an object at a further object- you can't have them both in focus at the same time. This is because vision is two dimensional. Depth exists, but perception of it is technically illusory.
This results in an instinctive preference for 2D media, which is what makes three-dimensional art difficult. The average brain simply isn't trained for three-dimensional imaging.
So imagine the difficulty of throwing a fourth dimension into the mix. It's difficult to even perceive.
But wait! There's a way! You see, if you hold a three dimensional object in front of a light, it casts a shadow. That shadow is purely two-dimensional, disregarding variations in the quality of the surface cast upon; the shadow has no depth, it cannot pool. In the z-axis, it exists at a pure zero mark.
This is how we are able to represent three dimensions in two- we observe depth as a shadow of sorts. The still images sensed by the brain are nothing more than the reflections of light from things- from instant to instant, depth does not exist. It's something like the argument about the arrow, wherein every instant the arrow is not moving but in real time it is moving swiftly.
It can be theorized, then, that a four dimensional object would cast a three-dimensional shadow. You wouldn't be able to manipulate it, per se, just as you cannot manipulate a 2D shadow except by moving the surface it is cast upon. It is possible that a 4D object could only cast a shadow upon the surface of another 4D object; I am unable to perceive whether I have any, and therefore cannot test my theory. However, the shadow can be made in three dimensional media.
My prevailing theory regarding 4D spaces follows this general line of thought.
Let's start with a 2D coordinate plane. We can add a third dimension by bending the plane; Let's say it's a window screen. Placing a rock in the center warps the screen, curving it downward. Now, looking at the way that the wires bend around the warping, you can see what the addition of depth does. Some of the holes between wires are stretched outwards; while we know obviously they have changed in area because we're working with a 3D media, were it 2D the surface area would have vastly increased while the circumference stayed the same.
Try and envision it now in three dimensions. The surface area is untouched, while the volume ramps up drastically. You could compare it to a bag of holding from D&D.
I can wrap my mind around it, sorta, but as soon as I try to figure out where in the 3D space your coordinates would be, I get a headache.