Some animals have no gills when young, but then develop gills that grow larger as the animal grows larger. what is the reason for this increase in gill size?

Young animals are very small, the oxygen easily diffuse inside their bodies, plus due to the small size they need little amount of oxygen to maintain body functioning. Larger animals need more oxygen to live and simple diffusion is no longer sufficient. Gills extend body surface in highly vesseled regions and that makes the diffusion a lot more efficient.