The basic story here is that:
a) when you burn hydrocarbons, you get CO2 and H2O and energy, plus some impurities and pollutants if your combustion process is imperfect.
b) they are "unburning" it by taking CO2 and H2O, adding energy, and making hydrocarbons which
c) can then be burned to get the energy back out.
This is NOT an energy source, but a means of storage -- you have to generate the energy some other way, then use it to make the hydrocarbons, then burn the hydrocarbons, and you're back where you started, minus any inefficiencies. Same idea (and result) as using hydrogen fuel -- you put the energy in to break hydrogen from water, and burn it back into water. You do not create or gain energy this way; you simply can generate in one place and burn it in another.
So what this does is to make CO2 a cycle, instead of creating new CO2 from fossil fuels, but you still need to get the energy from somewhere.
It could be of use if coupled to a nuclear reactor, or solar/wind/renewables, using that energy to make the jet fuel, and then burning it as you please.
PhilB