I believe the Mg+ Ca+ have a stronger attraction to the Cl- ions in the salt so when soaked in brine they trade places with the Na+

Beads prefer Mg over Na but Mg prefers Cl over beads. So you can recharge with salt brine.

Or it might just be a concentration thing where Na / Mg ratio wants to be the same in the brine and beads. So you just give a really strong brine and most of the Mg etc leaves the beads.