What is the chemistry behind this "petrol from air" technology?
Step one: trap $\ce{CO2}$. This can be done using specially selected amines, like triethanolamine. Aminosilicones (a common chemical used in hair conditioner), can also do this.

Step two: release trapped $\ce{CO2}$. In case of triethanolamine you'll have to boil mixture.

Step three: mix $\ce{CO2}$ with hydrogen (hydrogen is produced by electrolysis of water). Volume proportion should be somewhere 1:3. 

Step four: threat the mixture with proper catalyst. This step may be divided into two stages: reverse water-shift reaction  and some variant of Fischer–Tropsch process with some type of alcohol synthesis as very interesting candidate. There are some multi-functional catalysts that can perform both stages at once.

Another variation on step 4 is the Sabatier process, which involves generating methane from $\ce{CO2}$ and hydrogen.  Typically the catalyst is Rainy nickel (powder).

The required energy in form of electrical current is probably from 1.5 to 3 times of heat of combustion of resulting fuel, which is in turn at least twice more that can be converted into actual work again.