Why are we unable to visualize higher dimensional space? Is there any special feature or structure in the visuo-spatial area of our brain that limits our perception of the world in only 3-dimensions? This is a question for neurobiologists.
I believe the greatest limiting factor is the capacity of visual memory. See the attached publication. That said, I would argue that the ability for individuals to accomplish mathematical problem-solving (often thought to involve visuospatial neural systems) in higher dimensional space indicates the capacity for something approximating high dimension visualization.
For your entertainment, on a more handwavy-fun level, I have often wondered if facilitating more robust visualization across a greater time scale might foster an individual's capacity for high-dimensional reasoning in general. Perhaps providing time-delayed visual stimuli as a tertiary source of visual information might augment the visual system allowing accurate observation across a greater timescale, rather than relying on rapidly decaying neural representations of scenes. The thinking here is that one visual source allows two-dimensional observation; triangulation via data from two sources provides three-dimensional stimuli; perhaps three sources would allow more accurate four-dimensional perception. That may then facilitate higher-dimensional visualizations and reasoning in general. Let me stress, these are the things I think about while I'm pipetting and picking worms; strictly hand-wavy.
Article Capacity limit of visual STM in human posterior parietal cortex
In an embodied cognition perspective, it can also be explained by the fact that all our sensorimotor interactions take place in a 3D space. As a child, we learn about our environment through the coupling between our actions and their sensory consequences. It can be assumed that our higher level cognition (i.e., thinking and imagining) is grounded in these low level sensorimotor contingencies.
In the same line, the following paper proposes that "The emergence of spatial notions does not necessitate the existence of real physical space, but only requires the presence of sensorimotor invariants called 'compensable' sensory changes.". These compensable sensory changes correspond to the changes in our perception that we can compensate by some bodily actions. According to the authors, this could be the key to understanding the notion of space.
Article Space as an Invention of Active Agents
Since the space around us is 3D then why is it that biological organisms with vision would have developed capacities to see in 4D? If another Universe exist that has a 4D space and life with vision then I am convinced that these 4D animals will see in 4D and imagine in 4D. But since we can only imagine with our 3D vision system then we can't imagine in 4D. It is mathematically very easy to conceive space in any number of dimension including infinite numbers but we cannot imagine objects that have more than 3D because we use our vision system for imagining objects. We can imagine 3D slice of objects of higher dimensions and doing so is usefull.
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