Conclusions about tesseracts, and about four dimensional phenomena generally, can be reasoned by analogy. Because of the limits of our experience and senses, however, they can never be envisioned perfectly.
A good primer for thinking about higher dimensions is the short novel Flatland by George Abbott Abbott, a 19th Century British mathematician. This fantasy work (which is readily available in large bookstores in editions by various publishers)describes a universe where people only perceive two dimensions.
People and other objects there are a bit like spills of liquid on a flat table top. Their world is “sliced” so thin that people there have no perception that that they are physically removed from the surface they glide on, or that they, or or anything else in their universe has any dimension behind length and width. A sphere passes through Flatland one day, and is perceived as a circle which continually changes in size.
Some grasp of how four dimensional phenomena would behave in our three dimensional universe can be gotten by making Flatland analogies and considering the behavior and appearance of three dimensional objects there.
Another very good book is The Fourth Dimension by Rudy Rucker, which is a good nontechnical discussion of the subject.
For some reason speculation about a fourth dimension of space seems to have especially popular in The United States and Great Britain. In addition to the writings of Abbott Abbott, there was even a discussion of the subject over a couple of issues in the 1880s of St. Nicholas, a popular American children’s magazine.
And then there was Charles Hinton, a mathematician from England who taught at Princeton around the turn of the last century. He is said to have probably been able to picture and understand the existence of a fourth spacial dimension better than anyone else in history. A thorough-going eccentric, he developed a memory and visual imagery system which involved picturing objects as being occupied by arranged cubes to which identifying names were assigned. Using his method he was able to memorize exactly how objects were constructed, and to estimate their volume instantly. A man who studied Hinton’s system later said that he had condemned himself to a living hell.
Also of interest is a very entertaining story by Robert Heinlein called And He Built a Crooked House. In it a visionary architect tries incorporating theories about the fourth dimension into the design of a radically new house. He succeeds better than he intended, and he and the new owners have no end of trouble after they get stuck in the house after an earthquake.
While I don’t have much grasp of what exactly is under discussion, I have read that many physicists believe that the existence of additional spacial dimensions are not only possible but probable. Lately there has been talk that the span of some of these dimensions, at least as they intersect with the known universe, may be a measurable, if very small, interval. So nearly as I can follow, that could mean we are living in our own version of Flatland, occupying and passing through dimensions to which we are oblivious