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Dont bother trying to visualize the thing, it cant be done. What you can do is see a 2-D 'shadow' of it, and that is what is depicted in the clip.
A tesseract is just a 4-D cube (also known as a hypercube). A 2-D cube is a square, a 3-D cube is a cube and a 4-D cube is a tesseract. It is a mathematical construct where each corner has 4 lines that come off of it, and each line is at a right angle to the other lines. A square has two lines coming off of the corner, and they are at right angles to each other. A cube has three lines coming off of the corner and they are at right angles to each other. A tesseract has four lines coming off of the corner and they are all at right angles to each other.
If you ask some specific quesitons I may be able to help you. It takes a bit of time to develop a sense about these things. There is a popular clip out there called "Imagining the 10th dimension" - I would not suggest that, its crap. A cool, short book that explores the idea of higher dimensions is "Flatland". Its old, and funny but also illuminating. Carl Sagan summarized some of the concepts in this clip,