What is the significance of the degrees of freedom (n-p) in the SER formula?

The degrees of freedom (n-p) represent the number of independent pieces of information available to estimate the population variance. Subtracting the number of predictors (p) from the number of observations (n) accounts for the fact that each predictor used in the model reduces the number of independent pieces of information. Using (n-p) provides an unbiased estimate of the population variance.