It comes from some study in the 60s or 70s showing that Chinese peasant children who were malnourished and had to carry heavy loads, e.g. water buckets, had stunted growth.
The key there is malnourishment.
Somehow this morphed into the meme that "lifting weights stunts growth!" This was debunked in articles in Muscle & Fitness, for one; I don't have the issues anymore, but on the flip side, the fact that NOBODY has a reference for "stunted growth" should be rather telling.
In fact, there is not even a proposed mechanism whereby muscular loads would close bone growth plates. Even female gymnasts, who come pretty close to the textbook definition of being malnourished and overworked, usually grow too big to continue. For the most part only the genetically teensy stay small enough long enough to make it.
Muscular workloads do not close growth plates. It doesn't matter if you are lifting "heavy" or "light". The key is whether you have adequate nutrition BEFORE the window closes on your longitudinal growth, so that during your "growth spurts" there is enough material there to incorporate into new growth.
In fact, strenuous exercise spikes HGH, and combined with the increase in appetite, would likely produce MORE growth (assuming plenty of food) than seen in sedentary children.
EDIT: I just went looking for sources for this, and I cannot find any. Unless someone has better Google-fu than me, take what I said above as an extremely suspect anecdote. It seems that the major risks are that coaches will push children too hard, causing damage to muscles, tendons and so on.