Well, that claim would be based on the differences in brain power. What would mediate their awareness of the larger world or of the ultimate nature of reality, apart from a powerful cerebral cortex which is present only in higher mammals and perhaps elephants and dolphins? There are degrees of intelligence in animal species corresponding to differences in animal neurology.
If we posit an immediate intuition of the nature of reality that all animals share, I'm afraid that's the claim that wouldn't be based on empirical evidence. What's the evidence that all animals are aware of and understand the nature of the universe?