The answers to this are complex. Becoming overweight is nearly 100% related to insulin and insulin resistance with the number of calories being of negligible effect in real-world scenarios.
In short, kids get fat the the same reasons that adults do. They eat too many carbs and boost their blood sugar levels. The body responds by releasing insulin into the blood to address the glucose situation (as glucose is toxic in it free-floating form in large amounts) and insulin works to bring blood sugar levels down by shuttling the glucose into the cells.
Of course, free glucose in the cells is not any less toxic, so the cells begin to close the "windows" through which the insulin inserts the blood sugar. This is known as insulin resistance.
When this happens, the cells in the body which will get this overload of sugar will be the ones that are LEAST prone to closing these insulin-windows, the ones that are the LEAST insulin resistant. Can you guess which cells these are?
The fat cells of course.
Once insulin resistance has been realized, the body will begin racking up fat quickly. Children of insulin resistant (fat) parents are even more at risk. Why? Because in the womb the baby is exposed to the mother's blood sugar landscape. Since the blood sugar is high for the mother, the baby responds by quickly developing and abundance of insulin-producing cells in its pancreas in the womb. The result is that the kid in these circumstances is born ready to pump out LOTS of insulin and is more able to quickly develop insulin resistance of his own unless carb intake is severely restrcited.
Of course, many of these people come from families on the lower income levels who tend to feast on cheap, carb-saturated foods like pizza, cereal, and McDonald's instead of steak and salad for every meal.
The result is this. Mom is fat because she's poor or ignorant (i.e. she doesn't know that calories alone don't make one fat but the consumption of carbs does) and so she has a bad womb enviornment for her kid who is predisposed to becoming fat once s/he's born and then is pidgeon-holed into becoming fat because of the food that is provided while growing - cheap food.