how to understand the sentence

Hi emilywell, this sounds very much like a phrase used by Londoners (Cockneys).

In London, some people us the word "Dogs" as a slang term for feet.

To say that they are "barking" means that the person is suffering from tired or sore feet, because they have walked for such a long time.

Example...."I have walked for eight hours and my dogs are barking. When I get home, I'll soak them in cool water."