Stir means move, or to mix thoroughly. Stir is a verb commonly used with spoons (so spoons stir, or mix drinks thoroughly for example). The person thinks they are a spoon. "Sit there and don't stir" means "sit there and don't move". This is a play on the word stir.
Doctor , doctor , I keep thinking I ' m a spoon . - Sit there and don ' t stir . Next .