I admit that I might be dumb, but, how does one look at atoms and say "My God! There must be many worlds than just our one?"

Well, we looked at atoms and found out that the only meaningful way to describe them is with quantum mechanics. This is the most precise and possibly best tested physical theory ever developed. And it says that if an atom starts out in state A, it will then naturally evolve into a state A+B.

Now, A and B are mutually exclusive. So what does that mean? One reasonable way to view it is that it is indeed physically in both states A and B as the theory says. That's ultimately what leads to the many worlds interpretation. The atom is both in state A and state B, and the universe accepts both of the different trajectories of reality that leads to.

This view is equivalent to a number of other ways of view things, all of which lead to the same prediction of physical behaviour for now, so essentially you can just pick your favourite.