The Blue Carrot is considered to be part of the Secret Adventure Mode (SAM) feature. Its effects are many, and how to get it is one of these things you'll just have to find out by reading the Secret Adventure Mode article. One of the more known effects is that it can make norns dance, while it makes the Communist Ettins act quite savage. It is also said that if you use the blue carrot, the SAM autosave is disabled. Rumors say that there are other coloured carrots in the SAM too.
The Solution: To find out more about the Blue Carrot read David's solution.