Just to be clear, white space is the safe zone that appears when you create an outpost or search/clear an area completely.
The most important part about "white space" in State of Decay is the fact that you can exit a car safely within them. In comparison, if you exit your vehicle in the middle of nowhere without white space on the map, then several zombies will charge you (regardless of their distance) and usually a few will spawn as well. The total is random, but usually you will end up with all nearby zombies that would hear the car normally plus at least two others. Specials appear to have a longer range for detecting the car outside of a white zone as well.
White space also effects how zombies react to your guns. In a zombie siege, firing your shotgun or revolver will not attract nearby enemies. This makes the shotgun or revolver the ultimate anti siege weapon. Combine the noise cancellation with the fact that there are natural choke points and you have an impressive defense for very little influence.
Firing outside of a white zone will cause zombies to spawn and charge you as well as direct other existing zombies to run towards you from all directions. In later breakdown levels, this restricts the usefulness of more powerful weapons that can't be silenced, since you literally could fire forever and not kill the spawning zombies. Each shot would kill a few and spawn more, so it makes no sense for looting, outpost building, and most missions to use a shotgun. (Except of course, in sieges).
You can use the white space to your advantage both inside and out of it. There are many places on the map where you can drive into a shed and instantly search it to turn it white without leaving the vehicle. Once the area is safe, you can exit your car without having nearby zombies or spawned zombies attacking you. This is great for traveling around to grab oil in farm sheds. Simply drive in, make the area white, drive away to despawn the zombies that heard you drive by, and then return to find an empty shed and possibly nearby house.
Making the zombies run to you isn't always a bad thing. Many times I will drive a car to a spot near a target location I want to loot, get out, and then sprint to that location. I might have to kill a handful of zombies max, but the majority will be drawn to my car. Then I can loot freely and either pick them off with a silenced gun or make noise to get them off the car. If you do use a noise, don't panic when they seem to not hear it and just sit on the car. Eventually they will also run off.