The persistent visibility setting is used for layers whose visibility can be changed by a “parent” layer. In this case, when a parent layer is turned off, then child layers are also turned off. The persistent visibility setting determines what happens when the parent is turned on again.
Create some layers that have sublayers. Then turn off the parent layer. Notice how the parent layer’s lightbulb is gray but the sublayer’s lightbulb is half-gray. This indicates that the layer will be turned on when the parent layer is turned on. Now, if you click on the half-gray lightbulb, it will change to completely gray. When you toggle between half-gray and completely gray, you are toggling a layer’s persistent visibility.