If you're going to try to use solar power to reduce the concentration of CO2 in the atmosphere, the only endpoints that make much sense are carbon fiber and diamond. Things like limestone require a counterion like calcium or magnesium for each captured carbon atom, which is hard to come by.
Routes to carbon fiber and diamond pass through methane, which is also the input for the vast majority of petrochemical processes (including at-use-point generation of hydrogen via steam reforming), so the main thing to do is set up industrial scale solar-powered methane plants that use water and atmospheric CO2 as their feedstocks, generating natural gas which can then be piped or shipped as LNG to where they're needed as either an energy source for electricity generation (in which case the carbon returns to the atmosphere as CO2) or as a synthetic feedstock for everything from methanol to dyes to plastics and, carbon fiber and diamond as long-term stable storage products (with uses in say construction etc.). Diamond Age, here we come.
As far as the rationale for using solar/wind electricity to do this, that should be obvious, you're converting an intermittent/seasonal power source into stored chemical energy, just as biological photosynthesis does. That stored energy can then be used as needed, during winter months and so on. Of course batteries make more sense for storing solar power for use at night (hourly), but chemical fuels are better for months-long storage or for long-distance transport of energy, e.g. you can make methane in North Africa / Middle East and then ship it to Finland in the winter.
If you really wanted to, you could also run this process with electricity from nuclear power but the way technology is going, using wind/solar is going to be about 10X cheaper at the inputs end. Regardless, this approach would allow for the complete elimination of fossil fuels from the energy mix, which is the only plausible way to stop global warming.