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?
Not for me, e.g. "remember, remember the fifth of November" is how we remember the date of Guy Fawkes Night in the UK. "Fourth of July", "14th of February", "First of April", etc.
I guess you mean in the States, but perhaps they say it that way because they write their dates M-D-Y.