She decided to kill the 14-month-old dog she was training to hunt birds when it attacked chickens, an understandable mistake for a young dog IMO, and bit her once when she grabbed it's collar (no word on how seriously). She killed a goat by shooting it twice because she didn't like it, calling it mean and smelly. Apparently she's also recently put down 3 horses for unspecified reasons.
She deliberately included it in her new book "to illustrate her willingness, in politics as well as in South Dakota life, to do anything 'difficult, messy and ugly' if it simply needs to be done." When criticized she said, "I guess if I were a better politician I wouldn’t tell the story here.”
Sometimes animals do need to be euthanized, but it's rarely a first choice. Animal lovers and those who hunt with dogs have not been kind in their appraisals of the decisions. It's just a meme and maybe I don't need to worry, but her talk about carrying that attitude into politics puts me off.
Finally, Trump is old, unhealthy and there's fears he's showing signs of dementia. So this time VP choice might really matter a few years down the road.
This is dogwhistling that she is ready to start executing people. Socialists, LGBT, and so on. She wants people who see SCOTUS deliberating on the presidential immunity to put two and two together. "Elect me and I will execute the dangerous animals."