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.
So we say we know evolution is an alien god, which can do absolutely horrifying things to creatures. And surely we are aware that includes us, but how exactly does one internalize something like that? Something so at odds with default cultural intuitions. It may be just my mood tonight, but this short entry on the West Hunter (thanks Glados) blog really grabbed my attention and in a few short paragraphs on a hypothesis regarding the Hobbits of Flores utterly changed how I grok Eliezer's old post.
I have to break here to note that was the most awesome fact I have learned in some time.
That last sentence just struck me with utter horror.