How did this spider evolve to mimic exactly a human face and arms?
I think this really is just a case of what you refer to in your question as anthropomorphism. We are very, very good at seeing faces. Here are some more examples.

Addendum

This site has pictures of quite a few peacock spiders, and as far as I can tell these are all the same species (The only peacock spider mentioned on Wikipedia is Maratus volans). 

The remarkable thing about these images is that individual spiders seem to have very different patterns on thorax and abdomen, and many of these are much less like human faces. This seems to me to weaken the case for some sort of mimicry.

Interesting fact: seeing faces and other forms where they do not exist is a psychological phenomenon called pareidolia.