The connection between childhood antibiotics and obesity is the gut microbiome, which is the collection of bacteria inhabiting the gut. Each person has a unique microbiome, and it changes with age and environment. The role of the microbiome in a host of diseases is a very hot research topic.
Infants are thought to acquire their starter microbiome from their parents, mostly mom. The presence of these organisms plays a very important role in gut development. Animals raised in complete sterility, so they have no microbiome, do not develop the same level of gut functioning that non-sterile animals do. We can’t do that experiment on humans, but it has been replicated in flies, mice, and pigs, so it is expected to hold for humans as well.
When we take antibiotics, we completely disrupt our microbiome. In adults, it returns to its previous state (or a very close approximation) relatively quickly. In children, it takes longer to recover. If that disturbance occurs at the right time in development, it could mimic the impact of a microbiome-free development, resulting in poor gut function. When you consider the role of hormones in coordinating gut function, metabolism, and appetite, it’s not a big leap to see how this could lead to obesity.
This is the theory underlying the cited paper. Much of the links are conjecture at this point, but all of the biological parts are intact and the links are reasonable.
If you are interested in the role of the microbiome in obesity, here are a couple of interesting reads. Try not to get too bogged down in the sequencing details. The biology is more interesting than the technology.
Comparison of microbiomes of lean versus obese twins, role of genetics
Obesity-associated microbiome in adults
Both links go to the HTML version of the articles. PDFs are available from there.