We have two objects, $x$ and $y$. Let there be an equivalence relation $R$ between them, such that $xRy$. How does this imply that their equivalence classes, $[[x]]_R$ and $[[y]]_R$, are equal?
If $a \in [[x]]_R$, then $xRa$. We know that $xRy$ and $R$ is symmetric, hence $yRx$. By transitivity, we get $yRa$. Thus $a \in [[y]]_R$. Now just exchange the roles.