The odds of two random elements of a group commuting is the number of conjugacy classes of the group

$$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$

If this number exceeds 5/8, the group is Abelian (I forget which groups realize this bound).

Is there a character-theoretic proof of this fact? What is a generalization of this result... maybe it's a result about semisimple-algebras rather than groups?