My entry here does not answer the question per se; however, I share with you an approach that I came across in a book several decades back. I recall reading this exceptionally good book on higher dimensions when I was a kid, though unfortunately I can recall neither the book’s title nor its author.
The book was so good in talking of higher dimensions because the author employed the analogy of 2-d creatures, very similar to coin-like beings, moving about in their own 2-dimensional space with a human observer of their world — which provided an additional insight into this question to the reader because now you could imagine the same question from a higher perspective. The 2-dimensional beings grappled with the same conundrum that we 3-d ones do: is there a higher dimension beyond the ones that we find ourselves in? And then the author went a step further: these 2-dimensional beings discover a 1-dimensional world, very similar to needle-like creatures moving about in a single dimension.
Going further, you can imagine the havoc it would create in their 2-dimensional world if a human decided to pick up one such coin-like creature from his simple world: it would have the same effect on his fellow 2-d beings as it would on us if we saw a fellow-human suddenly disappear into thin air. Though, of course, there would be nothing supernatural in this for the 3-dimensional perpetrator of this act.
I read this book more than 30 years back, and unfortunately I do not recall all the details. However, it did fire my imagination — I was a youngster back then with no mundane considerations like making a living, so I had plenty of time on hand to be able to do that 🙂 — and IIRC I actually visualized a higher 4th dimension when I woke up in the morning after a particularly intense thought session the previous night wrestling with this challenge, though I am not so sure now of the exact thing that I had visualized; you see, a lot many summers have passed since then.
That said, I would ask the OP to employ the same approach by moving backward and forward between 1-d, 2-d, 3-d, and higher dimensional worlds; possibly that may do the trick for you.
Original post: https://www.quora.com/How-do-I-visualize-higher-dimensions/answer/Najeeb-Shaikh