Very simply -- Vomit.
Have you ever seen a dog eating vomit?
Usually refers to the vomit resulting from a full night's drinking, hence, being so drunk, one vomits in the street, and, being that drunk, one has usually acted very stupid throughout the evening. The phrase "A dog's breakfast" is usually attributed to the way one has acted prior to the resulting vomiting. Phrase popularized by the sit-com Kath and Kim (AUS).
Sharon was totally out of control at my bachelorette party -- she really made a dog's breakfast of herself!