Elemental chemistry
The chemistry to make fuel from CO2 isn't especially hard - split water into hydrogen and oxygen through electrolysis, add the hydrogen to CO2 to make carbon monoxide and water, then bung in more hydrogen to build up hydrocarbon chains.
This last bit's called the Fischer-Tropsch process, and dates back to the 1920s.
But it's the technologies capturing the CO2 straight from the air that are new and now becoming cheap enough to be viable.