That's not a dumb question; I've been thinking about this all day and getting depressed over how much quantum mechanics I've forgotten.
The question the paper asks isn't so much whether the wavefunction is "real" or not, but of how to interpret it. The question the paper attacks is this: Is the wavefunction a representation of physical property of a system ("real")? Or is it a representation of the information we have about the system ("mathematical tool")?
An analogy: I show you a coin. You measure its diameter and find that it is 24 mm across. That number--24--represents a physical property of the coin. Now, I flip the coin and cover it with my hand. You observe that the quarter has a 50% chance of being face up. That number--50--is not really a property of the quarter, but a property of your knowledge about the quarter. To prove this, I can look at the quarter while keeping it hidden from you. From your point of view, the probability of heads is 50%, but for me, it is either 0% or 100%.*
The paper is a proposed proof that the wavefunction of a system is more akin to the diameter of a quarter than to the probability of it being heads up. In short, the wavefunction is a property of a system, not of our knowledge of a system. Given a complete physical description of a system, there must be exactly one wavefunction that corresponds to it. Thus, the wavefunction is the physical description.
* There's an argument to be had here between Frequentists and Bayesians which I am totally uninterested in.