Refers to the rather unfortunate tan you get on your arms from wearing t-shirt's in the summer. Very similar to 'Drivers Arm', the Farmers Arm originates from the country region of Australia and can also be obtained from driving down long dirt roads in the summer with 3 other mates and a carton of beer.
Katie: "Let's go to the pool, its so bloody hot"
Linda: "Yeah, I need to get rid of these farmers arms"
Linda: "No way, I'm not going anywhere with this farmers arm"
Linda: "OMG! I've got farmers arm"