"How did you see this?" Dumbledore asked quietly, still not looking at Harry.All the other times I have read this book I assumed that Voldemort was possessing the snake (Nagini) and as Harry had access to his thoughts and feelings, he was also inside the snake. This time, however, I wondered if perhaps, because a bit of Voldemort's soul was in Nagini as well as in Harry, he was accessing Nagini's view of the Department of Mysteries while Voldemort was elsewhere? Perhaps because they were both Horcruxes, somehow Voldemort could communicate with/control Nagini, and Harry could tap into Nagini's mind as well as Voldemort's?
"Well . . . I don't know," said Harry, rather angrily---what did it matter? "Inside my head, I suppose---"
"You misunderstand me," said Dumbledore, still in the same calm tone. "I mean . . . can you remember---er---where you were positioned as you watched this attack happen? Were you perhaps standing beside the victim, or else looking down on the scene from above?"
This was such a curious question that Harry gaped at Dumbledore; it was almost as though he knew . . .
"I was the snake," he said. "I saw it all from the snake's point of view. . . ."
I might be completely wrong, but the Riddle in the diary was able to function on his own as he possessed Ginny Weasley, so perhaps the pieces of Voldemort's soul in the various Horcruxes could connect to each other, particularly the two in living beings.