It doesn't make any sense to me. Like he says at the start, he's supposed to be describing spacial dimensions. But then he uses time, quantum wavefunctions, and all sorts of other weird things which are not space to represent higher dimensions. If you think about the three spacial dimensions we live in, an important attribute is that they don't have an order. That is, if you choose an x, y, and z axis, there's nothing special about them. You could swap the x and y, for instance; they both behave in just the same way and describe the same kind of thing. In fact you can point your axes in any directions you like (as long as they're not in the same plane), and you can then uniquely describe a point in space using the distance along each axis (which is the definition of a space having three dimensions).
Algebraically, multiple (flat) dimensions are extremely easy to 'visualise' and work with. An example of a point in three dimensions is (2, 7, 5). An example of a point in ten dimensions is (5, 8, 3, 5, 1, 9, 3, 5, 8, 3). If I went to that point and then moved 100 units in the 6th dimension, I would get to point (5, 8, 3, 5, 1, 109, 3, 5, 8, 3), and so on. That's all there is to it.
Drawing a four dimensional cube is also pretty easy. I recommend you try it yourself before you read the following, it's a really fun puzzle. The key is to try to find some kind of pattern that takes you from 0D (a point) to 1D (a line) to 2D (a square) to 3d (a cube).
Another way to think about it is to go back to coordinates. A square is described by the points (0,0), (1,0), (0,1), (1,1). A cube is each of these four points, with a 3rd coordinate which can be either 0 or 1. So a four dimensional cube is described by points (0,0,0,0), (0,0,0,1), (0,0,1,0), ... , (1, 1, 1, 0), (1, 1, 1, 1).
String theory is complicated by some of the dimensions being curved, but I'm not knowledgeable enough to tell you much about that, except to recommend ignoring anything the video said about the subject.


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