Question

how does minimizing within cluster variation imply that the clusters are at maximum distance

Answer 1

K-means is all about the analysis-of-variance paradigm. ANOVA - both uni- and multivariate - is based on the fact that the sum of squared deviations about the grand centroid is comprised of such scatter about the group centroids and the scatter of those centroids about the grand one: SStotal=SSwithin+SSbetween. So, if SSwithin is minimized then SSbetween is maximized.