Why is "It was a dark and stormy night.." not a good opening?

Also, why has it spawned an award for bad writing?

Answer

It's an unsubtle cheat.

(That doesn't necessarily mean it's not good. Bear with me.)

The author wants to get across that Really Important Things are Happening. He wants to hook you with the beginning of his book. How does he manage to impart the tremendous significance the reader should be seeing right from the start?

Answer: by giving a dramatic, atmospheric, visual. That'll be enough to give readers a striking opening, to get across that Big Stuff Is Happening.

It's a cheat, in this case, because the visual really has nothing to do with the story. It's powerful, but irrelevant - the author is invoking the visual just to draw you in, not because it fits the story or is actually significant.

And it's unsubtle because he hits you over the head with it. "It was a dark and stormy night" - that's just a hair away from writing "It was a dramatic, suspenseful night," which I think you'll agree would be absurd. The author's trying to tell you the story is significant, and doing so quite bluntly, instead of actually demonstrating its interest and significance.

Now, in moderation, those would be legitimate cheats. You can definitely get away with that kind of stuff; often it's to your advantage, as long as the reader doesn't notice outright what you've done. The difficulty is, "Dark and stormy night" is both such an egregious example, and so oft-quoted and overused, that it completely loses its effectiveness on the one hand, and calls fatal attention to its flaws on the other. It's both a cliche (because a stormy night is an effective image) and a useless one (as opposed to cliches which have true power because their core idea is so significant).

Others have responded regarding the contest. I don't think that the tiny snippet is the inspiration for the contest; the full paragraph and Bulwer-Lytton's prose in general are to blame for that.