Unfortunately, for most developers there is no use. The official guide for android:persistent has a weakly worded comment that it is "intended" for system apps. To be clear, this property is ignored unless you are a system app.
You can see in the commit where flag checking was added to PackageManager, this requires both the persistent flag and the system app flag. If your app only has one of these, it will not be treated as persistent.
So this property is only useful for platform developers and hardware vendors. Sorry.
But what does it do? This property allows you to start a background service on Oreo and prevents it from being automatically killed.