When does the standard errors of OLS estimates decreases when we have more explanatory variables? So far my understanding is that standard errors of $\hat\beta_j$ increase when we have more independent variables in the regression model because variation among those independent variables will increase ($\rightarrow R^2_j \uparrow$). But I saw the opposite thing happened when I add more explanatory variables in the RHS of the simple difference-in-differences model:
$$y_t=\beta_0+\delta_0x_1+\beta_1 x_2+\delta_1x_1x_2+u $$
I can't understand how this could be possible?

There are competing by factors. You’ve identified the variance inflation factor (VIF), and you are right that adding more variables can increase the VIF. However, another factor is how large the residual variance is. After all, a major factor in the standard of your parameter estimates is the residual variance.
Consequently, the standard error of your parameter estimates decreases when the decrease in residual variance overwhelms whatever increases to VIF occur due to adding in new features.
