At first, it may seem that going beyond three dimensions is pointless, because many might think that if we want to describe something in our concrete physical world, certainly three dimensions (or possibly four dimensions if we consider time) will be enough. But in reality, it isn't. I often run into high dimensional vectors in neuroscience, and there are many other examples where records of data sets can be represented by high-dimensional vectors in high-dimensional space, for example the position of an airplane in space (it's a well-known example): one would think that it would take three dimensions to do that, one each to specify the x-coordinate, y-coordinate, and the z-coordinate of the airplane. Although it is correct that one needs only three dimensions to specify, for example, the center (or any given point) of the airplane, it is however important to note that the airplane could "rotate". In fact, it can rotate in three different directions, such as the roll, the pitch, and the yaw. Consequently, we would need six dimensions to specify the precise position of the airplane: three to specify the location of the center (or any given point) of the airplane and three other to specify the direction in which the airplane is pointing. Your vector representing the position of the airplane would then be in six dimensions.
In fact, you can view them as "rows of numbers", but they could mean something. They could physically represent something, even if it's hard to visualize. For visual representation of n-dimensional spaces, you could search about high-dimensional spheres and cubes and consider the representation of their vertices.
