In general, the relationship is similar to that between belief that Jerusalem fell in 586bc and the lack of belief in the JW doctrine. Evolution is a fact that goes directly against fundamentalist Christianity (and Judaism and Islam) and therefore it helps many to see that if one doctrine if flawed then others may be too.
It also removes the necessity for god in yet another realm, much the way Laplace did when he proved that the solar system didn't need a divine being to keep it stable. The more knowledge we gain about the natural processes that have gotten the universe to where it is today, the less necessary it is to appeal to a deity to explain things. As people learn to find natural explanations to things, they cast off superstitions, including belief in god.