HAR1 speedy mutation
Substitution rate at neutral sequences

The rate of substitutions at neutral sequences is given by the mutation rate. It si a very simple and classic result. It is because the probability of each new neutral mutation to fix is $\frac{1}{2N}$ (which you can calulate from a Moran branching process as well as from a simple Wright-Fisher model of genetic drift) and there are $2N\mu$, new mutations at each generation (where $N$ is the population size and $\mu$ the haploid mutation rate), resulting in a substitution rate of $2N\mu\frac{1}{2N} = \mu$.

Substitution rate at non-neutral sequences

A beneficial mutation has a probability higher than $\frac{1}{2N}$ to fix, while a deleterious mutation has a probability lower than $\frac{1}{2N}$ to fix. Therefore, the more beneficial mutations there are, the higher is the substitution rate. There are a number of approximations to the fixation probability of a non-neutral mutation. For example, using diffusion equations, one can approximate the probability of fixation of a deleterious mutation with selection coefficient $s$ by $\frac{1-e^{-\frac{4s}{N}} }{1-e^{-4Ns}}$ from some Kimura paper.

This rate of substitution therefore depends upon the selection scenario. It is lower than $\mu$ in conserved sequences and it is high than $\mu$ in sequences that undergo positive selection. So, the reason for such fast evolving sequences is just that they are under positive selection.

How about HAR1?

Without looking for more accurate statement form the scientific literature, the same wikipedia article you cite says


  HAR1A is active in the developing human brain between the 7th and 18th gestational weeks. It is found in the dorsal telencephalon in fetuses. In adult humans, it is found throughout the cerebellum and forebrain; it is also found in the testes.[1] There is evidence that HAR1 is repressed by REST in individuals with Huntington's disease, perhaps contributing to the neurodegeneration associated with the disease


It already gives you a vague idea of potential selection pressures on those sequences in the human genome.