The best putters tend to be players with very little body movement during their putting stroke. They putt from their shoulders with a rocking motion that sees the left shoulder lower and the right shoulder higher on the back swing (right handed golfers) and vice versa on the follow through.
This creates a pendulum stroke to the putting action and allows the body to remain parallel to the target line throughout the stroke.
Keeping the body parallel to the target line means that the putter head will be swung along the target line in a straight back straight through action and with the putter head remaining on the target line throughout the stroke, if the putter face aims down the target, the ball will be accurately struck straight along the target line. As a result of this, you will hole more putts and reduce your scores.
A no-peek putting stroke means that you are keeping your eyes very still throughout the putting action. Your eyes look at the ball as you make your back swing and then connect with the ball. Then as the ball begins to travel along the target line and towards the hole, in a no-peek putting stroke your eyes remain still and do not follow the moving ball. They stay in position and now look at the ground where the ball just was. Doing this maintains your body alignment to be parallel to the target line and as a result, the putter head swings more directly along the target line, increasing your accuracy.
If you allow your eyes to follow the ball once it has been hit and is travelling along the target line, then your head will turn and as a result of this your body will also turn towards the target. This will result in the putter head being pulled off the target line to the inside of it, or left of the target and with the putter head travelling less along the target line, your accuracy will be lower and you will tend to miss your putts to the left if your putter face is also aiming left of target.
A great drill to work on to promote a no-peek putting stroke, is to putt with your eyes closed. Go to the practice green and set yourself up ready to putt. Take a look at the target and the target line and then look back at the ball. Now close your eyes and use your imagination to picture the target line and the green between the ball and the hole. If you cannot picture the line of the putt when you first try this, just open your eyes and have another look at what you are facing, then close your eyes again. Keep practising doing this and remembering how the situation looks so that you have a good memory of the target and the line of the putt and the more you practice this, the better you will get at visualizing. It will also help if you visualize the target line in a very vivid colour, such as bright red, against the green, as it will attract your attention to the line to hit the ball down.
Once you are able to close your eyes and picture the hole and the target line well, simply play your putt with your eyes closed. If your eyes are closed you will not turn your head to allow your eyes to follow the ball and as such you will not peek at where the ball has gone. Keep your eyes closed until the ball is at least half way to the hole and then open them. Once you have got used to doing this and playing the putt to the memory of the hole and target line, then putt with your eyes open once more and work on keeping your eye still and on the spot that the ball has been hit from, again until the ball is at least half way to the hole.
Working on this drill will really help you to maintain your body alignment, help you to play the putt from your memory and by visualizing and as a result, you will become a much more accurate putter. This will get you holing more putts and as a result you will see that you take fewer shots on the putting green and your score will come down.