As far as I can tell, the paper is asking this question: if the world is just a wavefunction, why do we see it as a bunch of material things? Tegmark is trying to show that viewing the world as a bunch of material things is somehow special, that it optimizes some physical or mathematical quantity. That's impressive if he can make it work, but I'm not sure it's on the right track. Maybe a better question would be, which ways of looking at the wavefunction are the most likely to contain evolution? After all, minds are optimized for the kind of information processing that is useful for evolution. (Um, what I really meant here was "useful for increasing fitness", thx Mark_Friedenbach.)
I think you're on the right track in assessing the paper's content. Here's what I retained from a first reading: He considers a quantum density matrix. He decides to separate it in a way which minimizes the mutual information of the two parts, hoping that this might be the amount of conscious information present, but it always turns out to be less than a bit. Also, his method of division tends to produce parts which are static (energy eigenstates). So in dividing up the density matrix, he adds a second condition (alongside "minimize the mutual informa...
Max Tegmark publishes a preprint of a paper arguing from physical principles that consciousness is “what information processing feels like from the inside,” a position I've previously articulated on lesswrong. It's a very physics-rich paper, but here's the most accessable description I was able to find within it:
The whole paper is very rich, and worth a read.