"People remember unfinished or interrupted tasks better than completed tasks."
When you leave tasks unfinished, they tend to stick in your head. This hurts your ability to focus on current tasks.
The getting things done system is designed to combat this. When you put a task in your inbox, you can consider it "taken care of." Therefore you are taking the cognitive load out of your head and moving it elsewhere.
While these systems help you deal with the Zeigarnik effect, they don't actually help with productivity. Writing things down feels like getting things done, but its really a form of self soothing. Hundreds of Ways to Get S#!+ Done—and We Still Don’t - Wired