According to statistics in the U.S. (women) initiate divorce filings 66% or 2/3rds of the time.
My guess is people who file for divorce after the first of the year see it as having a brand new start for themselves. It's possible they may have gone back and forth over whether or not to stay or move on. Symbolically a new year represents a "clean slate".
People are more courageous or optimistic about changes at the start of a new year.
A new year also elicits hope for a better year than the previous one. I've heard "being together" (more) for some couples causes problems. I've even read reports of people breaking up after taking vacations together or filing for divorce immediately after their honeymoon! Holiday seasons oftentimes puts people together longer than they're use to.