They kill off prey, so the predator population decreases and the count of prey increases again. It is in equilibrium. Nature has defended herself. That is the Gaia concept, that Nature defends herself. Another example is that there are definitely bad effects from burning out the Amazonian forests, but the effects are more muted than expected.
Fewer forests mean more carbon dioxide in the air.
Forests live on carbon dioxide. They love it and it helps them grow. Forests grow thicker all by themselves and turn carbon dioxide into oxygen. Killing off forests fosters growth in other forests and Nature has once again defended itself. So the Gaia people are partially correct.