Are there any deep details on the specifics of what the “Low Power Mode” in iOS 9 actually affects?
Apple hasn't really discussed specifically what Low Power Mode actually does. Its main underdocumented feature is that it underclocks the processor, reducing performance to increase battery life, but there's a few other tweaks it makes to the system as noted underneath the switch.

As shown in these screenshots, Low Power Mode reduces the performance by ~30%.


  


Source: http://www.macrumors.com/2015/06/25/ios-9-low-power-mode-benchmarks/

Apple lists the following as features modified by Low Power Mode:


  
  Email fetch
  Hey Siri
  Background app refresh
  Automatic downloads
  Wi-Fi associations
  Some visual effects
  


Source: https://support.apple.com/kb/HT205234

More details are provided in the developer documentation (Energy Efficiency Guide for iOS Apps):


  
  Reduce CPU and GPU performance
  Pause discretionary and background activities, including networking
  Reduce screen brightness
  Reduce the timeout for auto-locking the device
  Disable Mail fetch
  Disable motion effects
  Disable animated wallpapers
  


Source: https://developer.apple.com/library/prerelease/ios/documentation/Performance/Conceptual/EnergyGuide-iOS/LowPowerMode.html

Apple encourages developers to take advantage of this feature by implementing a lower power state in their apps. This means that this feature may increase battery life through support in third-party apps.


  Your app should take additional steps to help the system save energy when Low Power Mode is active. For example, your app could reduce the use of animations, lower frame rates, stop location updates, disable syncs and backups, and so on.