YYYYMMDD orders dates the same way you orders numbers: most significant portion first. MMDDYYYY would be like writing "one hundred twenty three" as "twenty and one hundred three". When storing numbers for machine use, there are significant advantages to keeping the ordering from most significant to least.
In our culture, we have a natural understanding of MMDDYYYY because, as humans, we have an awareness of time, and years progress slowly. We generally know what year it is. Seeing the year rarely matters, so we push it to the back. Months change over just fast enough to retain their importance. Other cultures handle this differnently. Much of the world prefer DDMMYYYY.