How does the act of producing animal feed from the leftovers of a process result in a negative carbon footprint?
Its only marketing, not a science equation... It's saying that their gin gives back CO$_2$ from other gin which uses fertiliser, and saves a bit of crop space for livestock food. Overall, any crops tend deprive a region of carbon sinks and biodiversity.
Cropland has reduced the worlds total CO$_2$ sinks by 50+ percent, so using 1 crop for 2 things saves CO$_2$ sinks, fertilizer and fossil fuel. Animal stock is intrinsically a climate change accelerator too, especially mammal stocks, which occupy forest space and produce methane and use forest space for hay bales and food crops. Saving surplus foods slightly reduces their impact, its reduces the CO$_2$ budget compared to non surplus feeds.