Why must gymnasts begin to train at an early age?

Flexibility is the key and it is much easier to train flexibility while the body is supple. hope this helps.

At a young age, a child (or even a baby when it's born) are already born very flexible and have a tendency to stretch more freely than people older that you.
This is why it's easier to mend the body and start to get it used to stretching at such a young age because the body will get used to it, and adapt more to it's flexibility, which will make it easier on the person in the future and as they grow up