Even at the end of the article the authors conclude:
> “The most plausible explanation for the totality of evidence on this is that those who are given PPIs, and especially those who continue on them long-term, tend to be sicker in a variety of ways than those for whom they are not prescribed.”
That's because this is an observational study. That means there is probably another confounding factor that is perpetuating the use of a PPI in some people, but not in others. Sometimes these confounding factors are controllable - as long as you know what they are - but usually they are not. For example, those who were taking omprezole may have peptic ulcer disease - which would put you at risk of gastric cancer, and for which the PPI is used to reduce the acidity of the stomach, and reduce the risk of ulcers and inflammation (which can be a precursor or risk factor for cancer). Similar story for gastro oesophageal reflux disease, where the risk is to the oesophagus.
If you want to answer the question 'Does taking a PPI increase my risk of gastric cancer?' you need another type of study. That is a randomised controlled trial, where you randomise people to either the PPI tablet, or a placebo, then measure what happens. This allows you to infer causality, and the randomisation prevents confounding factors.
Thankfully this has been done[1]. It's only one paper from a low-impact journal, but none-the-less a meta-analysis of several RCTs setting out to answer a specific question holds more weight than an observational study.
Also, I'm not a gastroenterologist, so if there is one out there who understands the field better, please chime in. Specifically, if you have a clear medical reason to be taking a PPI (i.e. reflux with oesophageal histology changes), are there RCTs comparing PPI use versus not? I've done a quick google and this [2] study seems relevant - they suggest that if you have mild disease you can just take the PPI when you have symptoms, but not for severe disease. However, they didn't assess cancer risk.
[1] https://www.ncbi.nlm.nih.gov/pubmed/23906249
[2] http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2036.2007....