If someone is shirty, they are in a bad mood and they show it: short answers, a rude tone, ready to snap at you. It is informal British English. You often hear don't get shirty with me when one person thinks the other is being unnecessarily sharp. Though actually, it has nothing to do with the shirt you are wearing. It is probably built on an old phrase, get one's shirt out, meaning to lose your temper — the opposite of keep your shirt on, which means stay calm. Americans almost never say shirty; they would say cranky, snappy, or irritable instead.