I knew it was a long time ago. Maybe he's afraid of hurting it again, or like I am guessing, it makes it easier for him to push and defend with some added support, like a brace to keep from folding.
I think he does it for support, like others have mention here, but I also saw somewhere where he said he does it because it's kind of something he's known for now. It's a very JJ thing to do. Dude onows how to market himself. I'm fine with it. It's his iconic look.
I imagine he is afraid of what will happen if he doesn't use it like how Harden got addicted to wearing the big patch across his shoulder for what seemed like years.