What does the battery saver do in PokemonGo?

Solution 1:

8/8/2016 update has added the Battery Saver backed into the iOS devices

Resolved issues with the battery saver mode on iOS and re-enabled the feature

In the 3/8/2016 update for iOS, the Battery Feature has been removed for the time being. The feature is still available for Android devices

We have been hearing the feedback loud and clear about the removal of the Battery Saver mode on iOS devices. We had removed this feature as several users were experiencing it to be buggy, but we are fairly confident that a fix for this should roll out within the next several days.

Support PokemonGo - Settings explains what Battery Saver does. Other than disabling the display when your device is downward, it doesn't seem to have any other purpose.

Battery Saver

Toggles the battery life optimization feature. Battery Saver disables the display while the device is pointed downward. Pokémon GO will continue to track distance traveled and notify you of nearby Pokémon while Battery Saver is activated.

Solution 2:

On some Android devices it turns the screen off when the phone is in your pocket. That doesn't work on other devices (and iOS doesn't let apps do that!) so it dims the screen if possible and stops rendering graphics.