I would like to know why Naruto came back with a cape to fight Pain? What is the meaning of the cape?
1 Answer
Although there is no canonical name, I would refer it as Sage Cloak. There is no meaning of the cape. It was used for aesthetic purpose (looked cool with the orange aura) and doesn't hold any chakra (he didn't lose the outfit when he returned to normal mode).
Many were disappointed as to why Naruto didn't keep the cloak after the Pain fight.
References
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On a side note, the author even went through the trouble of showing the destruction of the cloak during the 6-tail Kyuubi transformation at the end of chapter 437.– Gae. S.Commented Nov 5 at 6:43