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?

This can happen for a few reasons.  One scenario is as follows:
Consider a model
$$ y = \beta_0 + \beta_1x + \beta_2 w + \beta_3 z + \epsilon $$
Suppose that the covariate $x$ is randomly assigned to units in the population.  Hence $x$ is uncorrelated with any other variables $w, z$ considered for adjustment by construction.
If $w$ and $z$ can explain variation in $y$, then the standard error for $\beta_1$ (the coefficient for the randomly allocated $x$) can decrease as compared to the model $y = \beta_0 + \beta_1 x$.
This happens because the estimate of the residual variation $\sigma$ will decreases (due to variation in from $w$ and $z$ explaining away variation in $y$) and the covariance for beta is a function of $\sigma$.
We can very easily see this if we just simulate the model
set.seed(0)
N <- 1000
x <- rbinom(N, 1, 0.5)
w <- rnorm(N)
z <- rnorm(N)
y <- 1 + x + w + z + rnorm(N, 0, 0.5)

# model without w, z
vcov(lm(y~x))['x', 'x']
#> [1] 0.009220088
# Look at the residual variance, nearly thrice as big as it should be
sd(resid(lm(y~x)))
#> [1] 1.516374


# Model with w, x
vcov(lm(y~x + w + z))['x', 'x']
#> [1] 0.0008677971
# Look at the residual variance, much closer to the true value
sd(resid(lm(y~x+w+z)))
#> [1] 0.4643699

Created on 2022-11-24 by the reprex package (v2.0.1)
