How does peculiar velocities lead to the observed elongation of galaxy clusters in redshift space, otherwise known as the Fingers of God effect? I have read the relevant Wikipedia page, but cannot understand exactly how it works.
Firstly, as you've mentioned, the apparent elongation happens in redshift space and not real space. What this means is that if you map the redshift of different galaxies in the galaxy cluster, the resulting shape will be seemingly elongated along our line of sight (the radial direction in the diagram below).
Also just so everyone is on the same page - A peculiar velocity of a galaxy is the velocity possessed by the galaxy in addition to the Hubble flow (i.e., due to the expansion of the universe). Therefore if one measures the redshift of a cluster of galaxies, we will get a mean value around which there will be a distribution of redshifts due to the peculiar velocities of these galaxies.
The galaxies will only show an additional redshift (of say $\pm \Delta R$) if there is a component of their velocity along our line of sight. Hence the only direction the galaxy can be seemingly elongated is radially (i.e., along our line of sight to the galaxy), which is why the galaxies seem to be "pointing" at us. Which led to the name. :)
This doesn't mean that we're sitting in a special place or anything. Any observer sitting anywhere in the universe will see a similar effect, for exactly the same reason as outlined above.