Question about fixed effects, and state-by -time fixed effects

Consider first the simpler regression: $$ y_c = beta x_c + mu_s + eta_c $$ In this regression, the idea is to use the variability of $x$ across counties within states to identify $beta$. Why would you want that? Maybe because other policies are implemented at the state level that might influence both $x$ and $y$. In this case, omitting the state fixed effects $mu$ would allow one to use both the cross-county and the cross-state variability of $x$, which might cause a bias.

Running this fixed-effect regression is exactly equivalent to centering your $y$ and your $x$ by removing the state average to each county-level measure and run the simple regression of the centered $y$ on the centered $x$. Or, as you put it, running one regression per state, and then average all state-specific estimates.

Now, let's go back to your specification: $$ y_{ct} = beta x_{ct} + lambda_{c} + mu_{st} + eta_{ct} $$

We are not only using the within-state variability in this case. For instance, if $x$ was varying across counties but constant over time, we could not identify $beta$. The presence of $lambda_c$ means that we are identifying $beta$ from the variability of $x$ over time and across counties within states.

This is a very flexible specification because it allows the different states to have arbitrarily different time evolutions in $y$, and counties to have arbitrarily different levels of $y$. For this reason, it is also quite demanding on the data, and you end up throwing away a lot of the variability in $x$, leading to less precise estimators.

Note that differentiating this equation over time leads to: $$ Delta y_{ct} = beta Delta x_{ct} + Delta mu_{st} + Delta eta_{ct} $$ with $Delta X_t$ defined as $X_t - X_{t-1}$. In differences, this model is therefore close to the first one: we rely on the within-state cross-county variability of the time difference of $x$.