Two subtractions do the work: each group’s own pre-period subtracts away its baseline level, and the control group’s change subtracts away shocks that hit everyone, leaving the treatment effect on the treated (ATT). In regression form it is the coefficient on the treated-times-post interaction, with standard errors clustered by unit.
The identifying assumption, parallel trends, says the treated group’s untreated trajectory would have matched the control group’s observed one. It is untestable but probe-able: flat pre-treatment trends and event-study plots support it, and a treatment triggered by a diverging trend violates it in exactly the way pre-testing misses. Staggered rollouts add a modern trap, since a single two-way fixed-effects regression mixes comparisons that use already-treated units as controls; heterogeneity-robust estimators exist for that case.