to grow up: to increase in size or to pass from childhood toward adulthood
This is the plain, physical sense of the word: something gets bigger, or someone gets older. It is what a child does over time and what a plant or a flock does as it thrives.
The two senses here are not really separate ideas, only the same growth seen in different things. A boy grows until he is weaned or old enough to act on his own; a tree or a herd grows until it is full and abundant. In every case the word simply tracks the steady increase from small to large.