I did bad in high school chemistry, I never could understand this stuff. First, the ion exchange resin beads are washed with Na+ which bind to the beads. Second, the hard water is run through the beads and bad stuff like Mg2+ has a stronger attraction to the beads than Na+, so the Na+ is pushed off and Mg2+ binds to beads. The resulting water out of tap is softer because Mg2+ is out, but saltier because Na+ is in. Then once in a while the beads are flushed with Na+ again with the Mg2+ being sent down the drain... but how does this happen if Na+ has a weaker bond? How can it push the Mg2+ off the beads??

The concentration matters. Concentrated Na will displace Mg, but in the Na concentration in the tap water Mg will displace Na. If you had a concentrated brine with large amounts of both sodium and magnesium, the Mg would win.

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.

As explained, it's the concentration. If you don't have a dishwasher with a water softener: You don't sprinkle a little salt in some compartment, it's large and filled with 1-2 pounds of salt to make a saturated brine. The salt sold for dishwashers is purified NaCl (table salt has some other stuff like anti-caking agent). That's what the dishwasher uses to drive the Ca2 and Mg2 off the resin.

Ca²⁺ is the more important ion, since limescale is mainly calcium carbonate (CaCO₃). Otherwise, the explanation is fine.

The exchange of the ions is driven be the difference in concentration between water and resin surface. It works both ways, but Ca²⁺ is preferred (=stronger bonds) by the binding groups (e.g. carboxylate salts). That's why a concentrated Na⁺-brine is needed to flush out the calcium ions.

Imagine the bonding as reaching an equilibrium where Mg2+ bonds and then dissociates in a repeating cycle. If you flood that solution with Na+, the chances for Mg2+ to bond after dissociating are lower when an excess of Na+ is doing the same thing, and a new equilibrium is reached.