This has multiple benefits. It encourages students to start working early instead of waiting until the last minute to start on their assignment. This means less stress for students and higher quality work (generally). It also benefits the teacher in allowing them to get started on grading early rather than waiting until a hundred essays come in at the same time and feeling totally overwhelmed and buried in grading.
This semester, as an experiment, I offered each of my classes an extra credit point on one of their big assignments if they turned it in a week before the due date. I wasn't sure how many, if any, of them would take me up on this offer, since I'm pretty sure they've never heard of anything like that being done before.
To my great surprise and pleasure, most of my undergraduate writing students turned their essay in early this week. Many of them even came out of their way to find me since their class this week is canceled (due to Dragon Boat Festival), and I told them if they wanted their extra credit point, they'd have to find me before or after one of my other classes.
My postgraduate students have an assignment due June 23. Same procedure for them; if they turn it in by June 16, they get one extra credit point. Today, the first student submitted her assignment. I haven't looked at it in detail yet, but from my quick read-through, it looks pretty good.