In Statistical Methods by Snedecor and Cochran, it states something to the effect of "the within-group variance is an estimate of overall variance given the null hypothesis"
and this point is used to justify that the ratio of between-group and within-group variances follows an F distribution under the null.
But why is this true only under the null? At first I thought it was some kind of weighted average, but with the quadratic numerator I'm not so sure.
In ANOVA the numerator of the F is the mean square for the within-group variation.
When $H_0$ is true, the variation in means is simply caused by the error term and so you can estimate the variance of the error from the variation in means.
When it's false the numerator consists of both that noise effect and the mean square difference in population means.
Indeed, that's the entire point of ANOVA - loosely, you detect a difference in population means when the mean square difference in sample means is larger than would be reasonably consistent with all population means being equal.
