Yes once a user leaves your site, it is pretty hard to know what happens on the affiliate site - you just have to trust the system they use. You can look at trends and your previous experience, but it is hard to have any proof.
I've never known an affiliate system to have independently verified stats. Whether you're an affiliate for a small business or Amazon, they control the stats and you just have to accept what they say they owe you.
He could've had someone test this directly, by clicking on the affiliate link and purchasing a product, ideally while recording their screen so there's hard evidence. Then if the company still claimed there were no conversions, he would know for a fact that they were lying.
As they mentioned further down in the article, they later got access to data on how affiliate references were performing and used that to extrapolate over the 120 cases that were missing data.