I don't know how this study does it, but one can make estimates by seeing how death-rates change in places with different levels of coal particulates, particularly over time as coal plants were opened and shuttered.

Alternatively there are studies that correlate various pollutants with respiratory issues, and then compare that with the pollution emitted by coal plants.

Ideally one would make many such estimates, yielding a range that one could have high confidence is close to the truth, similar to how we use multiple methods of dating fossils, and then know something is wrong if there is a large disparity.