Indicator Variables and Interactions
An indicator, or dummy, variable represents membership in a category:
\[D_i\in\{0,1\}.\]Difference in levels
In the model
\[Y_i=\beta_0+\beta_1D_i+\varepsilon_i,\]the conditional means are
\[E[Y\mid D=0]=\beta_0, \qquad E[Y\mid D=1]=\beta_0+\beta_1.\]Therefore,
\[E[Y\mid D=1]-E[Y\mid D=0]=\beta_1.\]If $D_i=W_i$ is treatment assignment in a two-arm experiment, OLS with an intercept gives
\[\hat\beta_1=\bar Y_1-\bar Y_0.\]This algebraic equality does not by itself make the contrast causal; causal interpretation comes from Randomized Assignment or another identification argument.
Interaction with a continuous variable
Consider
\[Y=\beta_0+\beta_1X+\beta_2D+\beta_3XD+\varepsilon.\]For $D=0$,
\[E[Y\mid X,D=0]=\beta_0+\beta_1X.\]For $D=1$,
\[E[Y\mid X,D=1] =(\beta_0+\beta_2)+(\beta_1+\beta_3)X.\]Thus $\beta_2$ is the group difference at $X=0$, and $\beta_3$ is the difference in slopes. Centering $X$ at a meaningful value makes $\beta_2$ easier to interpret.
Experimental interpretation
Treatment-covariate interactions allow the treatment effect to vary with baseline covariates. They are also used in fully interacted regression adjustment:
\[Y_i=\alpha+\tau W_i+X_i^\top\beta+W_iX_i^\top\gamma+\varepsilon_i.\]Common trap
When an interaction is present, the main effect of $D$ is not an overall treatment effect. It is the effect at the reference value $X=0$ unless variables were centered or the model was otherwise parameterized.