If you're looking for any reasoning in that movie, you're looking for the unattainable.
To answer your question, you might recall that Morgan Freeman explained that the magicians stole the money as it was being loaded on the truck. They replaced the money with flash paper, which ignites without leaving any residue or smoke. That flash paper is what ended up in France, while the money was brought back to the magicians. This is how they made it seem as though the money disappeared. Meanwhile, the person they pulled from the crowd was dropped into an identical looking vault below the stage, and the money was siphoned out of this "stage vault" and out into the crowd.
Of course, flash paper needs an ignitor to work. Which means there had to be some ignition source in the actual vault in France; some source of heat or flame to make the flash paper ignite. That part was never explained because it's impossible to do what the movie said they did. Not to mention how they transported that much cash back to America without even being questioned.