As I understand it, in the specific context of linear regression, the R output "residual standard error" is an estimate of $\sigma$, the standard deviation of the distribution of the residuals. You can compute $\sigma^2$ by the MSE, which in general is the mean of the squared errors, but in regression the denominator is the residual degrees of freedom $n-p$ ($p$ representing the number of parameters including the intercept).
$$MSE=\frac{\sum_{i=1}^{n} y_i - \hat{y}_i}{n-p}$$
Why isn't residual standard error just called RMSE, when $\sigma^2$ is the MSE? Or is $\hat{\sigma}^2$ often called the residual variance (making the terms for $\sigma^2$ and $\sigma$ line up)?