It's far more difficult for a prey animal to have its muscles/fat/organs suffused with chemicals that will kill/injure its likely predators, especially if its predators have similar metabolic pathways as itself -- it'd be at risk of poisoning itself if the poison broke free!
Animals certainly fight back against being eaten (running away, fighting, etc) but once they're dead their meat (barring like, snake venom glands or pufferfish or other poison/venom-suffused sea creatures) won't fight back.
Obviously "plants are poison" is phrased for shock value and there are traditions and safe handling practices (like "cooking", or "not eating green potatoes" https://en.wikipedia.org/wiki/Solanine) for plants that remove, denature, or prevent the formation of defensive chemicals.