Why do Americans want to know the month first and the day second?
I'm sorry but it doesn't make sense TO ME. Based on what I was taught, regardless of the month, I think what matters first is to know what day of the month you are in, if at the beginning, in the middle or at the end of said month. After you know that, you can find out the month to know where you are in the year.
What is the benefit of doing it the other way around?
I think it's just the way we talk. It's just more common for us to refer to a date in speech like "Today is June 1st". Whereas other countries would say "Today is the 1st of June". Neither is wrong, it's just how things are said.