It's an idiom: "If you tell someone you are going to see a man about a dog, it is a way of saying that you do not want to tell them where you are really going."
Now, let's use it in a sentence.
"Is Christ the Number One priority in your life?"
"Oh I tried going to church, but being a good person and acting religious with all the other hypocrites didn't help me at all. Besides, I don't believe in a God who says he loves everybody but still sends some people to hell. I'd love to explain how mainstream religion changed Christ's message and why everybody's path to salvation is different, but right now I've got to go see a man about a dog."
