Why can’t animals exist without plants?

For two main reasons:
First, animals do not have the ability to produce oxygen that they breathe. And for this reason, animals are dependent on plants. Therefore, large forests are called the lungs of the planet, because all of us (and animals) can breathe thanks to plants.
Secondly, plants are a food source for all animals. This even applies to meat-eating predators. After all, they hunt herbivores, and those without vegetation will die out. So indirectly, plants are food for everyone.