ANOVA is based on an F -ratio that is calculated as the ratio of two variance estimates, the variance between groups and the variance within groups,

ANOVA is based on an F-ratio that is calculated as the ratio of two variance estimates, the variance between groups and the variance within groups, but enables conclusions to be made about the means of the samples involved. What is the logic of that? I.e., explain the rationale that supports the use of variance estimates.