The argument is that this factor can minimize the mean squared error of the variance estimate, at the cost of some bias. In general, the small correction to the lead factor depends on the fourth moment. For a Gaussian, you get 1/(n+1).

This is an example of a “shrinkage estimator”, which comes up a lot - introduce some bias but get a smaller MSE. For more, see: https://en.wikipedia.org/wiki/Bessel%27s_correction