The energy comes from excess generation in renewables for load balancing, that base load thing people mistakenly say they can't do.

It's clever and simple, you put a whole load of potential generation in knowing that to meet your essential and desired demand on low generation days you'll need excess capacity which will over produce on high generation days. You then plug that in to a system which has tanks of feedstock in this case poo and empty storage capacity so that in peek generation periods it can run at maximum, when it's only a little over the requested load it runs at limited power and if there's a time with no excess power it turns off for a bit.

That's why all the carbon capture and processing facilities are focusing on modular parallel design, it's very easy then to create scalable production tied to excess load.

Of course this is only one of the many possibilities, the nuclear lovers want to build nuclear powered sequestration and processing facilities, Iceland made one using geothermal, the American one is wind and the proposed Saudi one trailer about being solar thermal.

Oh and actually the efficiency is incredibly impressive now, with some of the active catalyst chemistry they're developing we're getting into heat pump style efficiency gains and it'll looking more likely we'll be able to go below parity in cost per gallon Vs mined hydrocarbons.

I know it feels like people never explain the complex side of things but that's because journalists are bad at their jobs, there's whole organisations out there dedicated to this sort of planning and a lot of the stuff they talk about and work towards ia incredibly well thought out and sensible.