On the surface, this is a short, wry poem about a pet being called fat. But the speaker's real target isn't the cat's weight. It's the people who say it out loud, "right to her face," as if the cat can't understand them. The speaker suggests she can, or at least that she picks up on the tone.
The second line turns the insult around. Maybe the cat is fat because of the teasing itself, because being pushed down makes her feel "anything but skinny." That idea has a bit of dark humor, since it's a funny way to explain extra weight. But it also reads as a hint about people, not just pets. Anyone who has been made to feel bad about their body might hear themselves in it.
The mood is protective and a little sarcastic. The speaker sounds fond of the cat and quietly irritated at the critics, defending her with a joke that carries a real sting.