Why did the atomic bomb get invented?

Answer 1

Mainly because a few physicists who had recently escaped Nazi Germany believed that the Germans were working on one, and convinced the US government that it was necessary for the US to develop it first.

In reality, while there were a few brilliant physicists (and a somewhat larger number of moderately competent physicists) left in Germany by that time, for the most part the ones actually capable of doing the kind of work needed to create the bomb were not particularly sanguine about the concept of doing so for the Nazis. They mainly managed to convince the German leaders that it was either impossible or too difficult to be worth bothering with (not particularly difficult, as it turns out; Hitler was, for most of the war, not willing to expend resources on anything that didn't promise to produce clearly superior weapons in a relatively short time... six months or so), and got themselves put into positions of directing the minimal research that was conducted, where they largely steered it into avenues they knew wouldn't work.

I recommend the book Brighter than a Thousand Suns if you want a detailed history of the development of the atomic bomb.