Snape was Dumbledore's enemy at the time.
Snape said "Don't kill me" because at that point he was a Death Eater, while Dumbledore was the leader of the Order of the Phoenix. If the situations had been reversed - if Voldemort had met a lower-ranking member of the Order of the Phoenix - the outcome would most probably have involved the words "Avada" and "Kedavra" coming from Voldemort's lips. Of course Dumbledore would have been less quick to jump to the killing option, but it's questionable whether Snape would have understood this: hanging out with Death Eaters for so long, he might have forgotten how decent people behave.
Later on, of course, Snape turned to Dumbledore's side and became a member of the Order himself, and a spy in Voldemort's ranks. But at the time of this scene, he hadn't yet defected - in fact, his defection is in this scene - so he was merely a Death Eater, meeting the enemy leader.
The light comes from Dumbledore Apparating.
You're right: it is certainly a strange-looking effect! The best explanation I could think of was that it was meant to represent either Snape's passage through time from one memory to the next or Snape travelling to his rendezvous with Dumbledore (doesn't look quite like Apparition normally does in the films, but then Snape has means of transportation that most wizards don't, such as flying).
However, @Au101 came up with a much better idea: the light comes from Dumbledore Apparating. At the moment when it first appears (just after Voldemort's face disappears and the word "Severus" is spoken), it does look a bit like a tall white-bearded figure just coming into focus:
And when we get our first clear glimpse of Dumbledore, the light is just fading from his clothes and body. The fact that Snape appears to be travelling through a tunnel of light in the intervening moments is just an artefact of the way the camera moves: he's actually standing in one place as the light of Dumbledore's arrival bursts around him.
The scene in question is at 2:45 through this video (direct link