I do a lot of driving in the busy streets where its most likely to hit other cars, motorcyceles or even kids. You might not know who comes in front of you suddenly while driving and hits you.
Its very important to use your peripheral vision here to avoid such sudden suiciders. Here’s how i use my peripheral vision while driving.

Many drivers get hypnotized on the road while concentrating on the center of the road ahead. Sometimes it becomes so obvious for them to avoid looking at the sides of the road.
I agree that you should watch the road, but nothing would cause you from the front i suppose. So you must watch out the sides which bring trouble.

Look on the rear view mirror meanwhile

To properly use your peripheral vision so that you do not hit the things which come suddently on your path, is to first ensure your eyes and gaze; are relaxed. You relax your eyes by not fixing your sight at one fixed point or path on the road.

Secondly, you must scan the left and the right side of the road with quick eye movements. Meanwhile, you should also glance in your rear view mirror.
Follow these two and you will be able to relax your eyes and see better and watch out for sudden trouble hitting on you.