The zigzagged paper streamer things you mention are called "shide." They are in essence wards for purification. They are often seen with the rice straw rope they are attached to are called "shimenawa." Together they marks the boundary to something sacred (or rather mark the boundary between the sacred and that which is not) and can typically be found can be found on torii gates, around sacred trees and stones, etc. They are intended to keep impurities out and purify the space within.
As the same time, they can also be use to prevent, or rather seal the passage of gods. According to the Kojiki (a oral collection of myths forming the basis of the Shinto religion.), shimenawa was first used to prevent the sun goddess Amaterasu from re-entering a cave to save the world from eternal night.
In the Shin Sekai Yori anime, it's mentioned "evil spirits" and "monsters" are roam outside the town and any child who ventures out alone would suffer terribly. Hacchijoume is the shimenawa that encircles Kamisu's District 66, erecting a sacred barrier that protects the town from outside forces.