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.