"This "do" notation may seem mysterious, as it i..."

Then the counterfactual conditional $~$\\mathbb P(\\mathbf x | \\operatorname{do}(X\_j\=x\_j))$~$ is calculated via:

This "do" notation may seem mysterious, as it is not part of standard probability theory. However, as Pearl shows in Section 3.2.2 ("Interventions as Variables") of Causality, second edition, all of this can be treated as a notational convenience, as a causal model can be reduced to a certain kind of PGM with variables for interventions, and the "do" notation (or lack of a "do") can be considered to be a statement about the value of an intervention variable.

See also this Powerpoint deck.