People often report this koan with out explaining the surface meaning. Chinese at the time considered dogs nasty dirty animals, that hold the same position as pigs in the western mind. Pigs or cockroaches or those parasites that burrow into your eyes. So the surface question is can the most contemptible being become enlightened.
The surface meaning of the most common answer, Mu is Buddhist jargon, meaning emptiness, sunyata. In the Mahayana system, you are enlightened when you realize the truth of sunyata, i.e. that you have no self (no atman), everything changes over time and various other things that I don't have space for here.
In my opinion, koan practice is a practice. You think about a question and always are told that your answer is wrong. During this process, many people report flashes of insight where they subjectively feel that they have grasped the answer, i.e. satori or sudden enlightenment. Reasoning takes time, so sudden enlightenment can't be done in step by step reasoning.
In China, there was a huge multiyear (multi decade?) discussion on if enlightenment was a matter of following steps or if it was something that happened to you in a flash. In China and Japan, the sudden enlightenment proponents won.
Some modern critics say that especially in Japan with a new funerary and post mortem outlook, have moved onto a form of Zen where sunyata is a sudden realization that you do have an immortal self, which contradicts early Buddhist thought.