Get your goat
Make you annoyed or angry.
The dictionary definition of goat is 'a ruminant quadruped of the genus Capra'. What's that got to do with being angry? Given the meaning of 'get your goat', we might expect to find goat as a slang term meaning anger or annoyance. That meaning is recorded in the US book Life in Sing Sing, 1904, which goat is given as a slang term for anger.
The phrase originated in the US and the first entry in print that I can find comes from a fanciful story about a burst water pipe that was printed in the US newspaper The Stevens Point Daily Journal, May 1909:
"Wouldn't that get your goat? We'd been transferring the same water all night from the tub to the bowl and back again."
The expression took a few years to cross the Atlantic. The first non-US citation isn't found until 1924 in the English author John Galsworthy's story White Monkey, and even there it is clearly seen as a recent innovation:
"That had got the chairman's goat! - Got his goat? What expressions they used nowadays!"
The following year, The Times printed a piece in memory of the then recently deceased Friedrich Baedeker. This included a side-swipe at American tourists and uses the phrase as a typical piece of Americana:
"... goggled Americans whispering aloud, 'Wa-al Sadie, these durned three star things get my goat'!"
A commonly repeated story which purports to explain the phrase's origin is that goats were placed with racehorses to keep them calm. When ne'er-do-wells who wanted the horse to race badly removed it, i.e. they 'got someone's goat', the horse became unsettled and ran badly. That's just the sort of tale that gets the folk etymology juices running. Let's just say that there's no evidence to support that story.
See other phrases that were coined in the USA.