Your question can basically be extended to all studies related to statistics. In simple terms you are generally looking for a relationship (e.g. between an increased number of deaths and air pollution levels). Of course finding such a relationship ALONE doesn't tell you anything about cause and effect. Even if you are only looking at deaths from diseases known to be linked to air pollution, you have to come up with a way to show that these factors depend on each other - that they are correlated.
Usually, you look at some sort of control group that will have almost all the exact same characteristics as the test group, except obviously for the one you actually care about. The more similar the two groups are, the better of course - ideally if you could find two identical cities, with identical people - with the only difference being the air pollution, then you could show a cause and effect easily. Since this will be very difficult to achieve, you instead have to use some
sophisticated statistical methods to study correlation of your variables. You can start reading about those methods here: https://en.m.wikipedia.org/wiki/Correlation_and_dependence
So, as long as those studies you refer to follow common statistical procedures, which I will assume they did, then yes, you could staticstically show a relationship and argue that X-many more people died from an increased in air pollution.
One of the studies in the link you provided is referring to a Research Letter in Nature (one of the most highly regarded scientific journals out there): http://www.nature.com/nature/journal/v525/n7569/full/nature15371.html
[Lelieveld et.al] are using a:
global atmospheric chemistry model to investigate the link between
premature mortality and seven emission source categories
They are using a
sensitivity study that accounts for differential toxicity
They are focusing on
mortality related to PM2.5 and O3
estimate of overall health impact depending on assumptions regarding particle toxicity
So, basically, they build a global model that will be able to correlate higher particle toxicity values with the number of deaths in different regions. They also talk about a sensitivity study, which will test if any change of specific variables may have extreme effects on their model.
Our calculations of air pollution related mortality are based on the method of the global burden of disease [...] applying improved exposure response functions that more realistically account for health effects at very high PM2.5
Of course it is just a model and not the reality, so the accuracy of their results will depend on the accuracy of this model, which is explained in more detail here: http://www.ncbi.nlm.nih.gov/pubmed/23245609
Overall, this actually isn't so much of a direct study with X number of participants. Nobody actually collected health information from participants, but they are using previously generated information (from the World Health Organisation) on toxicity response the effects of air pollution onto a global level. I assume the WHO has collected thousands of samples and analysed clinical trial that do what I mentioned earlier - they study the effect of a particular change in your environment onto your health. So, by now knowing that O3 is actually actually reducing your life expectancy by X percent, they can make the aforementioned conclusions.