Unfortunately the research full report is behind a paywall, and the abstract/summary leaves out a lot of info. However the findings are interesting in linking the dysregulated eating behavior to childhood SES. The details of the full report are needed to clarify some questions, like the demographic composition of the sample populations.
It's been observed for a long time that some minority groups (e.g., black and Hispanic) have higher rates of obesity. The fact that these groups are also over-represented in lower SES population segments could be a factor in the results. It's not clear how such variables were accounted for.
Perhaps a key was the last experiment discussed in the article, where there was reduced response to blood glucose level. Ordinarily increased glucose acts as a feedback signal of nutrient status turning off the drive to keep eating.
Problems with satiety (sense of fullness) are common in obesity, though how it works is incredibly intricate and poorly understood. There are dozens of genetic elements and a vast array of hormones and other body messenger systems implicated, ultimately it involves most every part of the body. No wonder theorizing about causes of eating/metabolic disturbances is frustrating and unsatisfying.
I'll take a stab at it anyway. IMHO nutritional factors should be a major suspect. A big change in diet over the last few decades has been the substantially increased intake of sugars put into processed/convenience foods. Fructose in foods has increased the most and thought to play a prominent role in higher rates of obesity. It's conceivable that early life exposure to high levels of sugars could bring about enduring abnormalities in body systems responsible for metabolic regulation.