I am constantly amazed at the bizarreness of the underlying theme. Anyway, Zum Kuckuck is apparently a pun or play on words or something idiomatic that doesn't translate. I'm also not sure about the theme. This game is as much about birds as Bohnanza is about beans.
Birds just happen to be the images on the cards.
You deal out the cards from the 60 card deck. The top card of each pile is flipped up and each player then plays a card simultaneously and everyone reveals. The highest card played gets the lower of the two Egg cards. The second highest card played gets the remaining Egg. If you get an egg card and you already had one, only the most recent matters.