It is weird. Doing great mathematics is like writing great poetry in a language only a few hundred people can read. And yet, every once in a while a mathematical poem changes the world.
Yes, and it's often the case that, faced with such a "poem", the world changes without ever understanding the mechanism of the change, as with both relativity and quantum theories.
Cedric Villani (Fields Medal 2010) wrote a book about his work [1] where he tries to give the reader a hint of what it looks like to be a mathematician on a day to day basis. It's quite interesting, even though he does not try to explain his theory.