Half the Muggles killings back when You-Know-Who was in power were done for fun.
I figure that the word 'back' here is used as an adverb meaning "in the past". The sentence can be rephrased as: In the past when You-Know-Who was in power, half the Muggles killings were done for fun.
But I'm not quite sure if my understanding correct?
The word "back" in this case is being used in accordance with definition 1b in the Merriam-Webster's entry for "back" as an adverb:
in or into the past : backward in time
So you are correct; it is saying that when You-Know-Who was in power, which was in the past, half of the Muggle killings were done for fun. The exact way you rephrased it, though, may be slightly off on the nuances. Since you started off with "In the past" it might convey that you are speaking about the past in general, and then specifying a particular point in the past — namely, when You-Know-Who was in power. But the way the sentence was originally phrased it does not include any implications of the past in general; it is referring directly to the specific time when You-Know-Who was in power.
