Growth is started by nesting. Another player must invite you to their nest before you can hatch. Once you have hatched, you will start to grow in size, gaining size once a minute. Every time you reach 20% (0.2) more growth, your creature statistics (health, damage, speed, etc.) will update. Though it may seem obvious, a juvenile's statistics are not the same as an adult creature. For example, an adult Shantungosaurus has nearly 12,000 health, while a newborn baby has only about 100.
To check the current growth of your creature, press the "insert" button on your keyboard. Once you have reached full growth (0.6/0.6, 0.8/0.8 etc.), progress to the adult form of your creature, and you will continue grow from your current size to full adulthood (1/1).
Currently, Tyrannosaurus has a more advanced growth system, involving juvenile, sub-adult, and adult models. Currently, all other "baby" creatures are simply downscaled adult models; however, in the future, all creatures will have at least two different models for growth.