Good question! I found this for you.

In Western culture the White Stork is a symbol of childbirth. In Victorian times the details of human reproduction were difficult to approach, especially in reply to a younger child’s query of “Where did I come from?”; “The stork brought you to us” was the tactic used to avoid discussion of sex.
http://en.wikipedia.org/wiki/Stork

Because they’re the ones who bring the babies. Duh.

I heard somewhere that a baby stork’s cry sounds a lot like a human baby’s cry, which is why the legend began in the first place.

…you left out the cabbage patch part, AC. My dad told me they got me with Green Stamps.

Here is an article on the subject: Storks & Babies