What exactly do things with a spell battery do? examples: Shamsuk's Amulet, Namer's Ring, Wizard's Hat...

It increases the amount of spells you can cast. Without them, each spell takes away a certain amount from your stamina, (the first part of you life). when your stamina is gone, you can´t cast any more spells. The spell batteries let you cast spells without taking away from your life, which means if you give one person all the spell batteries, he/she won´t run down on life before the battle is over.

don't forget that a spell battery will recharge by camping (regain half) or staying at a inn (regain full battery)