Derived from the use of a carrot on string supported by a stick tied to the neck which is hanging a few inches in front of horse, donkey or mule. This is a way to entice the animal forward. The animal never actually gets to eat the carrot.
Implies that the animal, person or country is not too smart and ruled by base instincts or political expediency.
Often used incorrectly to describe enticement (the carrot) from the front and threats from the rear (the stick, for beating). Usually in a political context
The USA is using the carrot and stick approach to nuclear disarmament with North Korea, promising food, fuel and trade in exchange for closing down the nuclear weapons plants. The North Koreans have complied but the USA has not delivered the promised goods.
Incorrect usage but quite common in the media.
Obama is using the carrot and the stick approach with Israel, promising them new F-14 aircraft if they sit down at the table with the Palestinians and discuss a two state solution to the conflict and threatening them with a cut off of military aid if they do not negotiate in good faith.