This question arises out of the question Is Haskell worth learning?

Generally a few often repeated statements are made, about how Haskell improves your coding skills in other languages, and furthermore, this is because Haskell is stateless, and that's a good thing.

Why?

I've seen someone compare this to only typing with the left hand, or perhaps closing your eyes for a day and just relying on touch. Surely there is more to it than that?

Does it relate to hardware memory access, or something else which is a big performance gain?