Why does "brain drain" exist?

This is not a very specific question, but I'm just wondering out of curiosity. I've heard the term thrown around in various contexts. The concept of brain drain seems to be the movement of people from a lower economic status country or area to another, presumably better off area. Why does it get called "brain drain" though? Why do people from other countries leave their countries to live in the US? In the news lately, I've seen mention of Cuba. If the US is so terrible and so reviled in the rest of the world, why would people from other countries want to come and live in the US?