The weathering of alkaline silicates naturally draws down atmospheric CO2. It's a slow acid-base neutralization reaction. CO2 dissolved in water forms carbonic acid, which reacts with alkaline rocks. Calcium silicate plus carbon dioxide turns into calcium carbonate plus silicon dioxide. The reaction is spontaneous under ambient conditions on the Earth's surface, meaning that it is thermodynamically favorable. It doesn't require any added energy beyond that naturally present in the environment. The reverse reaction that separates carbon dioxide from calcium carbonate again is thermodynamically disfavored. It requires a large energy input, as in the making of quicklime from limestone for cement production.

The geological carbon cycle based on silicate weathering is what will naturally neutralize human CO2 emissions on a time scale of hundreds of thousands of years.

The reason it takes hundreds of thousands of years is that the chemical kinetics -- rate -- of the natural reaction are very slow, being limited by the available reactant surface area. This is the same reason that e.g. a steel hammer left outside in a rainy region takes years to completely disintegrate to rust, while steel wool under the same conditions will disintegrate to rust in under a year. The thermodynamics are the same in both situations: iron oxidizes spontaneously. But the kinetics are much faster when the material has a large surface area exposed.

Most of the exposed weatherable silicates on Earth are in the form of huge chunks: boulders, mountains, and region-spanning plateaus. The idea behind accelerated silicate weathering is to crush huge chunks of silicates down to sand-size particles so that the surface area and reaction rate increase dramatically. If the crushed material is dumped into shallow ocean water near shores, wave action also provides additional "free" mechanical grinding to further accelerate the process. Using these silicates to neutralize excess soil acidity on agricultural land, where limestone would normally be used, is another way to further accelerate the chemical transformation.

The human energy input required for accelerated silicate weathering is still large in absolute terms, but much smaller than trying to turn CO2 back into carbon and oxygen. It might take 5% of a coal plant's electricity output to crush enough silicates to offset its CO2 emissions. (Though ideally you would run the process on renewables, since crushing can be scheduled flexibly and only annual throughput really matters.) The process reverses ocean acidification effects of CO2 as well as reversing warming effects from CO2 in the atmosphere. It doesn't require artificially concentrating CO2 out of the atmosphere.

I believe that accelerated silicate weathering can bring atmospheric CO2 back below 300 ppm in less than 1000 years, though still more than 100 years. That's assuming that anthropogenic emission rates decline over time, mind you.

It makes sense that the vast majority of the discussion around AGW mitigation is still about cutting emissions. While emissions are still growing, even ambitious plans like large scale accelerated silicate weathering can't offset them. Still, if you look at the IPCC reports and other scholarly literature, scientists are looking ahead beyond emissions cuts. The term they use is "negative emissions."