I read chapters of it a few years back, too, but forgotten the details. All in all, I was pretty convinced of the soundness of the approach at the time.

My recollection is that this framework of causal inference allows one to ask questions about a probabilistic model that one can then try and measure to test causality.

These questions are interventions or assertions (the do operators) that something happened.

So one would start out with a Graphical Model like in the smoking example which defines a probability model, and then make do assertions on the model for various candidate causes and see what that would imply about the change in probabilities and then design an experiment to measure them and confirm them.