I agree that some of this stuff seems counterintuitive on the surface. Once you make the connection with high-dimensional Gaussians, it can become more "obvious": if Z is standard n-dimensional Gaussian random vector, i.e. one with iid N(0,1) coordinates, then normalizing Z by its norm, say W, gives a random vector U that is uniformly distributed on an n-Sphere. Moreover, U is independent of W --- this is related to the fact that the sample mean and variance are independent for a random sample from a Normal population --- and W^2 has Chi-squared distribution on n degrees of freedom. So for example a statement about concentration of volume of the n-Sphere about an equatorial slice is equivalent to a statement about the probability that the dot product between U and a fixed unit norm vector is close to 0, and that probability is easy to approximate using undergraduate-level probability theory.
Circling back to data: it is very easy to be mislead when working with high-dimensional data, i.e. data with many, many features.