I don't know all the details, so I was going to let someone else explain it. I noticed the button and the big spike in the projected exp gain, too. I was wary of pressing it, but I eventually tried it once and it said I lacked an item required to do it. I assume it's some consumable that you get as a gift or bonus from somewhere. In any case, I would avoid pressing it until you know where the item comes from and how rare they are.