нашел произношение данной фразы, заинтересовался и получил такое объяснение от UrbanDict:
Basic Definition: To annoy you to the point of getting pissed.
Sub Definition: Goat: The goat is a metaphor for your state of peacefulness. When your goat is with you, you are calm and collected. When your goat is stolen, you become angry and upset.
Notes: Getting someone's goat can not be a quick process and must be done by not being directly mean. The best way to get someone's goat is by means of clever annoyance.

в общем так и не понял про какую козу и спокойствие идет речь)

и примеры использования:
- I'm going to get your goat!
- I got Sam's goat today.

PS самое интерестное что ни тут
http://ru.forvo.com/search-en/get%20goat/

ни на
http://www.urbandictionary.com/define.php?term=get+goat

нету словосочетания "get goat"

Патомучто get someone's goat, также как и get up somebody's nose

To annoy or anger a person.

http://www.thefreedictionary.com/get+someone's+goat