It turns out that after seeing Goku using this power during the Majin Buu arc, Vegeta had been training in secret to reach the new form himself. And through the fight with the Tamagamis ...
As was explained by Bulma in the latest Daima episode, Vegeta apparently trained his way to the new form, doing so on his own which threw Goku for a loop. For Son, the transformation was learned ...