Words:鬼Oni = a devil /
のno = genitive joshi particle /
居ぬInu = absent; an archaic form of Inai: not be present /
間ma = during a certain period /
にni = ablative joshi particle /
洗濯Sentaku = laundry; washing clothese /
When the cat's away, the mice will play.
The "devil" here is about someone you fear or feel uncomfortable in their presence (e.g., boss or supervisor), and "laundry" suggests having a relaxed time or fun. Finally, the boss is going out. Let's do Oni no Inu-ma ni Sentaku!