The title and the short itself are a play on The Honeymooners.
There's been no food in the house since the people moved out, so the mice, Ralph Crumden and his friend Ned Morton, are happy when a new family moves in and brings groceries. They try to steal food from the kitchen, but are surprised to find it guarded by an orange cat.
They build a "Trojan dog" from a barrel and operate it from inside. They walk it over to the refrigerator, but while they are busy loading it up with food, the cat climbs inside of it and waits. When they get back inside, the cat attacks, the "dog" is splintered, and Ralph and Ed barely make it back to their mousehole alive. Ralph's wife Alice asks why they don't just get rid of the cat. Ralph says if she thinks it's so simple, then she should just do it herself, so she does.