"hung" is the past participle (過去分詞) of "hang".
"hang over" is to describe the crappy feeling after consuming a lot of alcohol the night before. You will feel nausea, headache, dizziness, and feel like all you want is to sleep the next day.
Hence, "hung-over tails" means "sleepy tails" (昏睡的尾巴). This is a more polite way of asking people to get up. Sometimes, in movie or daily life, people might even use "hung-over axx".