The basic idea is fairly simple. From quantum states (the wave function) we can determine probabilities of outcomes when these states are measured in different manners. A major problem in quantum foundations is how to understand these probabilities arise, i.e. where do these probabilities come from?
For example, it could be that the probabilities arise from our ignorance of knowledge of the system (these are what people traditionally call hidden variables.) Imagine a hard disk with a bunch of information on it. If you don't know all of the bits on this hard disk, then your computer can act in ways you can't explain because you don't know all of the bits. Each time you setup your system and run a program you may get different results because that extra information on the hard drive could be different, and so because you don't know all of the information you will see probabilities of outcomes. Mystery solved. Coming up with such a theory however currently always runs into a problem with locality. But that's a story for a different comment thread...
Here is what the authors ask. They say: well it could be that all of the information that is specified in a quantum wave function is all that matters, that is for each quantum state there is a one to one correspondence with a configuration of the hidden information (they allow for extra information, but that's basically irrelevant.) That is, literally, the quantum state is written on your hard drive in all its glory detail, and each quantum state is distinguished for any other quantum state. Contrary to this it could be that the information overlaps in some way. That is for different quantum states, some of the bits, say, will be in the same configuration. The authors then go on to show that this later assumption isn't compatible with the predictions of quantum theory (and show an experiment that can be done that will verify that the later interpretation is not correct. If the experiment fails, then quantum theory is wrong, and all hell will break lose. Assuming quantum theory holds, this shows the second interpretation isn't viable.) Very neat.
There are a couple places where the argument seems a bit odd to me. For example, it is really not clear to me why the measurement device in their system has to depend on the portion of the information that is shared between different prepared wave functions. If this information is ignored by the measuring device, I don't see how their contradiction will arise. Of course it self tells us something kind of interesting because it puts a limit on how shared information is revealed to a measurement device (this seems almost Kochen-Specker theorem like.)