LP5912-EP

I'm trying to understand how the LP5912-EP (or similar) LDO can be used
to extend battery life.

From the data sheet the input V must be 0.5V greater than the output V
For a 5V output the input must be 5.5V

  8.3 Power Supply Recommendations This device is designed to operate from an input supply voltage range of 1.6 V to 6.5 V. The input supply must be well regulated and free of spurious noise. To ensure that the LP5912-EP output voltage is well regulated and dynamic performance is optimum, the input supply must be at least VOUT + 0.5 V. 

For a 6V battery pack the regulator will no longer output 5V once the battery voltage goes below 5.5V.

How does this help to extend battery life?
The battery is far from depleted.