Monumental proof settles geometric Langlands conjecture

How do people even find the time to work on this stuff without being distracted by life, family, and everything else? I think this is why so many of these people are in Europe. America is too chaotic and full of obligations and distractions to do serious academic work.

By my count at least four of the researchers are employed by American universities and therefore most likely live somewhere in the United States.

And "this stuff" to which you refer is the intended output of their full time jobs*. So presumably, they find time to work on it in the same way a software developer finds time to write code. You just sit down and do it, because you are being paid to do it.

*Did I miss something about how these papers were developed in their spare time?