Reddit mentions of Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension

Lies My Teacher Told Me by James W. Loewen was an eye-opener for me. I read it years ago, and haven't read the updated version, but I did find that one interesting.

Also Michio Kaku's "Hyperspace" is thoroughly entertaining and educational.

Think of the dimensions as fundamental parameters, conditions that describe the underlying basic qualities of the particles. They're not properties, like density, so that description doesn't really properly explain them either.

Michiu Kaku, who catches a lot of crap for not dismissing some ideas which are pretty clearly woo, is nonetheless a really great theoretical physicist. His book Hyperspace gives some great explanation of higher dimensions than I can summarize here. There is an easily-Googleable PDF of the book available for free, but I recommend buying it and supporting the author and his work, and the willingness of publishers to commission scientific books for laymen like this one.

Reminds me of Plato's Allegory of the Cave and HyperSpace