"Begin with the end in mind," famously wrote Stephen Covey as part of his Seven Habits, thus winding the ending up into the beginning. Yet the beginning gets wound up in the ending, too: because "closure" allows us to learn from the past, put it behind us, and start fresh. There are really two states of being, then: the ending/beginning state (or transitions, or time-between-time) and the work state, "working out" what has begun until it is ended.
"Closure" is really a kind of measure of how successfully we transit from ending to the beginning: no matter how bad you failed at the working-out state, you can "successfully wrap things up" by achieving closure. It's a chance to yank victory from the jaws of defeat, success from the midst of failure: if you just learn from it, you can draw the curtain down and move on. You can't have a good beginning without a good ending.
Closure gives us a fresh state through learning from the lessons of the past. You can't just walk away and still 'achieve closure.'
There's a bit of the Gospel in this: an eschatological message (seeking closure on this life), a salvation message (the clean start), the daily living message (each day a new day).