I think it was used because the author's English wasn't that good. Because normally, you'd write "WWII" and not "the WWII" (see this NGRAM), and you could just say "At the end of WWII, the Cold War began".
BUT as others have pointed out, you could justify the past perfect tense with some context.
For example, if the author believed that the Cold War actually began before the end of WWII, they could say: "The day the war ended, I was looking for a way to get from Moscow to Paris but I had no passport and at the end of WWII, the Cold War had already begun, so there was no way for me to go."
Then they could separate the long sentence into shorter ones for added drama and there you go, your original sentence is explained and justified :)