The key insight is that you don't need to turn carbon dioxide back into carbon. You just need to store those carbon atoms it in a stable form that keeps them out of the atmosphere. The most plausible way of doing that at large scale is to accelerate the reaction of naturally occurring silicates from rocks with atmospheric CO2.

"From a thermodynamic point of view, inorganic carbonates represent a lower energy state than CO2; hence the carbonation reaction is exothermic..."

That's from chapter 7 of the IPCC Special Report on Carbon Dioxide Capture and Storage, which I recommend reading for more details.