The article mentions what they plan on doing with it:

"What Gebald and Wurzbacher really want to do is to pull vast amounts of CO₂ out of the atmosphere and bury it, forever, deep underground, and sell that service as an offset. Climeworks’s captured CO₂ has already been injected deep into rock formations beneath Iceland..."

This is the crux of why this won't work - although it gives great branding for a company in the NYT. First if you put it into tomatoes or soda -- it gets released once the product is used. So you get a slight offset from not producing more CO2 for those project types but it isn't a clear win. Also the cost of geological sequestration for CO2 is quite expensive even if you are co-located by geological formation that works.

Either way - progress that people are looking into these possibilities and that they are getting mainstream attention... let's hope we can as a society keep building on successes.