I think the trouble might come from imagining the process as a gradual process by which a dog population evolved into a tumor population (which is not what happened; the wording in the original post is pretty misleading). The dog-to-tumor part is actually the easier and less shocking part of the story. Tumors are basically just cells that by some mutation have trouble regulating cell division and then divide uncontrollably. Malignant tumors (what we call cancers) are just tumors that happen to harm the organism (and maybe metastasize). So this particular tumor was once a dog cell, just as every human cancer starts out as a human cell. The interesting part of the story is that the tumor got to have a limited ability to survive outside of the original dog's body, and got to be able to survive within other dogs and other canids.