It is an idiom which is used in informal contexts. You normally use it when you don’t want to tell someone where it is that you are actually going. So if you want to keep your destination a secret and don’t want to reveal the purpose of your visit, you say `see a man about a dog’.
Examples: “I don’t know where Arvind went. He said that he had to see a man about a dog.” * “Prabhu had been moody all day. He suddenly got up and said that he had to see a man about a dog.”
“This informal expression has another meaning as well. When you say that you are going to see a man about a dog, it means that you are going to use the bathroom. It’s an indirect way of saying that you wish to use the toilet.”
It’s like the expression that ladies often use, `powder my nose’.
When ladies say they wish to powder their nose, what they mean is that they wish to make use of the bathroom.”
Example: “Order a dosa for me. In the meanwhile, I have to see a man about a dog.”