That page is not completely correct. It should be " I have to _hand in_ my essay ". Not _hand out_. To "hand out" is to distribute for free, while to "hand in" is to submit.
I can see why they might have made the mistake, because professors routinely say "please hand in your essays by Monday", and "please hand out these flyers for me". It seems obvious that students should hand out their essays then. However, hand in is just a synonym for submit. The "in" in that sentence is not independent.