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Fleming has been appointed on a four-year deal that will mean he coaches England in the next two Ashes series, and Vaughan said managing director Rob Key has selected a "very experienced" option.
Former England Test coach Andy Flower was initially linked with the role, but Royal Challengers Bengaluru and London Spirit coach ruled himself out of the process.
Fleming joins after leaving Chennai Super Kings this month, having coached the Indian Premier League (IPL) franchise for the past nine seasons.
"Andy Flower, I think from what I've read is the one that they wanted. He turned that down so it was like 'OK, how do we get the next best?'" Vaughan said of Fleming.
"Some will say he's not coached in red-ball cricket but you've got a very experienced manager, very experienced coach who's been around the modern game with Chennai Super Kings over the course of the last few years.
"He was a very, very good captain back in the day, a very good tactician, renowned for being a really good man-manager.
"Having McCullum still as the white-ball coach and then having someone coaching the Test team who doesn't know McCullum, that could have been really tricky.
"I think Stephen Fleming has been brought in for that reason - he's obviously very close to McCullum.
"They needed someone who could work with McCullum because someone that doesn't know Baz and his style, I think it could have got very, very messy."
Fleming, who retired from playing in 2009, has a wealth of experience in Test cricket, making 111 appearances for New Zealand, scoring 7,172 runs with an average of 40.06 in the middle order.
McCullum was known for promoting an aggressive form of cricket, dubbed Bazball by fans and the media, but Vaughan said Fleming will have to make changes to England's style and technique.
"I'm kind of sick and tired of talking about England's style of play. The style of play I want to see is England winning," Vaughan added.
"Whatever that style is on that given day, session, hour, England have to play the right style of cricket that wins games of cricket.
"We can only talk about talent and the gulf of talent in English cricket for so long - they've got to start winning games of cricket and winning trophies now.
"The one thing this Test team needs is coaching the technical side of batting.
"How many of them can look themselves in the mirror and when Michael Neser, Matt Henry, Scott Boland bowls that good length just on or around that off stump, can play a controlled forward defence shot?
"They won't like saying it and they won't like talking about technique, but technique wins Test match cricket and the technical side of England's players and batting has been found wanting.
"So it's all right putting a good leader of the group, but I also think they need one or two changes in the backroom team to make sure these players are getting really well coached as well."
