Weight versus Height is not just for the BMI (Bullcrap Mass Index, I mean you can just look in the mirror for a better approximation of your body fat percentage).
First, does losing fat help gain height? Yes, a higher body fat percentage leads to reduced insulin sensitivity. Insulin stimulates cellular proliferation and differentiation. Stem cells proliferating and differentiating into chondrocytes into the growth plate is what causes height growth (or stem cells proliferation and differentiating into the osteoblasts in the irregular bone). On the other hand, leptin increases osteoblast proliferation. However, osteoblasts only help you gain height in the spine (ever notice how obese people have longer torsos than average?). You need to stimulate chondrocyte proliferation and differentiation in order to grow taller.
The second linkage involves stem cells. Stem cells can differentiate into adipocytes, chondrocytes, and osteoblasts. The more cells you have differentiating into chondrocytes and osteoblasts then the less stem cells are differentiating into adipocytes. Adipocytes forming adipose tissue and what causes fat.
Fat also produces estrogen but I don't believe that estrogen is involved as much in height growth as some.
The third linkage involves muscle. The more fat you have, the more muscle you need to carry around that fat. Muscle increases insulin sensitivity
Losing fat has another benefit in that it makes it a lot easier to load your bones. Loading the epiphysis of your long bones increases intermedullary pressure which helps deliver the stem cells into your growth plate. Even if you are post natural development then you can still grow taller by delivering stem cells into your hyaline cartilage growth plate line if those stem cells then differentiate into chondrocytes (they have recreated a growth plate using a mesenchymal stem cell like chondrocarsanoma).
They way of growing in the irregular bones is by impact and fat dampens the impact. Impact causes microfractures in the trabecular which releases MSCs which then differentiate in osteoblasts which proceed to deposit bone beneath the periosteum thereby increasing your height.
