Itachi states that by killing those important to you, you can achieve the Mangekyou Sharingan. Sasuke, in fact, develops the Mangekyou after killing Itachi, who was important even when Sasuke simply wanted revenge. Kakashi has stated that "all those important to me are dead," implying that he actually already has the prerequisites for achieving the Mangekyou. During the timeskip, he must have figured out how to release his Mangekyou based on already having lost those important to him.