When you get someone’s goat, you succeed in annoying the individual or in making him rather angry.
*Sometimes, Raja comes to work in jeans and a T-shirt. That really gets my goat. *Mohan’s constant whining gets my goat.
This idiom comes from the world of horse racing. I understand that horses are highly temperamental creatures. In order to keep their prized possession calm and relaxed, trainers often kept a goat as a stable companion for the horse. Goats do not get flustered very easily, and their presence in the stable had a calming influence on the horse. Since the goat was a constant companion, the horse became rather attached to the animal. In order to ensure that a particular horse didn’t perform well in a big race, stable boys working for rival owners sometimes stole the goat the night before the big event. The absence of the goat made the horse moody, and as a result it didn’t perform well in the race. So when you say that someone has got your goat, you are comparing yourself to the horse, and not the goat.