I can only answer for my situation, but I know in the case of my daughter it was to increase her height. They determined by genetics that she should be 5'4'' tall when she stops growing. At the rate she was actually growing she would likely not even reach 5'. She never had a weight problem, only height. The theory is that if they can increase her height to where she "should" be, it would expand her chest cavity and allow her lungs more room to grow and expand. That would help with her lung issues. That may not be very technical, but that's the layman's way of putting what I was told.
Once we started having all the problems, it was obviously not worth it to keep her on the growth hormone. (at least to me... the dr did ask if I wanted to put her back on at a lower dose and I said no way).