Why They Got Better!
There are three main reasons why the players who got looked at the the hole throughtout their stroke got better.
When a baseball pitcher stares at the catcher's mitt he's using his natural binocular vision. In sport s terms this means your ability to see your target using both eyes. Your brain translates the subtle differences in what each eye sees to give you highly accurate depth perception - in this case, the distance between you and the target - which you can more readily translate into a feel for the proper length stroke. At the same time, you also take advantage of your focal visual system which sees things like break and grain. When you address a putt looking at the ball, your brain has to remember all that stuff, and with each millisecond you linger over the ball; you're less likely to remember it because you've cut off the visual information feed.
You've heard it on TV a million times, "Look how still Tiger (or Ernie or whoever) stays over the ball." If your body moves even a little while you putt, you can't make consistent contact with the sweet spot. Maybe it was a fear of whiffing the putt, but testers who looked at the hole maintained their posture like statues.
In other words, they established natural speed control. The most common reason by far for bad putting is the inability to consistently control the length of putts. Call it speed, distance, pace, whatever - bad putting is almost always caused by putts struck too hard or too soft. When you lack confidence in your speed control, your brain is always screaming "Hit the brakes!" as you start toward the ball. Slowing down your stroke - decelerating through the ball - is a death move for any putt in terms of both distance and direction. (The face opens invariably when you decelerate.) This creates even more problems on long putts because of the length of the stroke required.