Ok, so a bit more research, I found out what the Battery Saver tickbox does in Pokemon Go (something that @scoobnz had told me about a month ago but I forgot!). It turns on the Pokemon Go screensaver when you turn the phone upside down.
I also found out why it’s not on by default: It’s buggy. The screensaver causes Pokemon Go to crash (application hang), which wasn’t a problem since it was still detecting Pokemon and clocking up kilometres on my eggs. I would simply have to restart Pokemon Go when I came across a Pokemon I wanted to catch.
The problem came when I hatched an egg while the app was frozen. I lost all the kilometres I’d clocked up in the egg and didn’t get the hatched Pokemon 🙁
I’ve since been using the application with the Battery Saver turned off. At least that way I won’t have any issues.