The way he explains the issue is a little roundabout, so I thought I'd try my hand at reproducing the basic argument in a more succinct manner. Note that this post assumes some prior knowledge of the Many Worlds Interpretation (MWI) of quantum physics and how it differs from interpretations that involve collapsing wavefunctions. Basically, if you understood the last sentence, you'll be good to go; otherwise the following might be a little jargon heavy. Nevertheless, I'll try to give background where appropriate. (If you're really lost, the best book to pick up is Feynman's QED, but if you're in a hurry, the Everett FAQ can answer basic questions with plain English. It's no substitute for reading Feynman, though.)
There are several possible interpretations of quantum theory. MWI explains just as well as any other, but it is often called out as violating Occam's razor. "Stipulating all these extra worlds", proponents of wavefunction collapse theories argue, "certainly counts as extra assumptions." Of course, there's no reason why nature must respect parsimony, just as there's no reason nature must respect induction. Nevertheless, both have proven effective at making correct predictions over time (pun intended), so a good response is needed.
Instead, the complexity referred to in Occam's razor has to do with the number of independent rules in the system. Once Hubble measured the distance to cepheid variables in other galaxies, physicists had to choose between a model where the laws of physics continue as before and a model where they added a new law saying Hubble's cepheid variables measurements don't apply. Obviously, the model with the fewer number of physical laws was preferable, given that both models fit the data.
Solomonoff inductive inference does it by defining a "possible program space" and giving preference to the shortest program that predicts observed data. Minimum message length improves the formalism by including both the data and the code in a message, and preferring the shortest message. Either way, what matters is the number of rules in the system, not the the number of objects those rules imply.
Theory-wise, MWI is nearly equivalent to other interpretations. MWI will predict what the Copenhagen interpretation will predict, so in terms of accurate predictions, they seem about on par. But while MWI stops there, most other interpretations start tacking on extra assumptions. (Except instrumentalism, of course, but it always wins the parsimony battle, and doesn't really count.)
All theories that posit a wavefunction collapse, for example, are positing something on top of the needed rules that agree with observations (pun intended). Why posit a wavefunction collapse whose only purpose seems to be a method of denying the reality of the wavefunction? Of course, maybe reality is such that the wavefunction does collapse. But Occam's razor would clearly prefer the option asserting fewer assumptions.
"But wait!", say the wavefunction collapsers. "MWI asserts way more stuff than any other theory ever made! Don't you see that the addition of a single simple law of wavefunction collapse does away with having to admit the existence of all the excess crap MWI forces us to believe in?"
Yet they are missing the point. If something is a deductive consequence of the rules of a system, then you get that extra thing for free without having to refer to Occam's razor. As Yudkowski puts it, if P(Y|X) ≈ 1, then P(X∧Y) ≈ P(X). All the "excess crap" that gets posited into existence with MWI is just the deductive consequence of the quantum theory all interpretations agree upon. Even in wavefunction collapse theories, many worlds exist until the collapse occurs. The extra rule positing a collapse is the real "excess crap".
Parable of the Invisible Spaceship
If you're still not convinced, consider Yudkowski's implied invisible spaceship.
"Suppose you're going to launch a spaceship, at nearly the speed of light, toward a faraway supercluster. By the time the spaceship gets there and sets up a colony, the universe's expansion will have accelerated too much for them to ever send a message back." They will be unable to interact with us even in principle, as they can never intercept our world-line. "Do you deem it worth the purely altruistic effort to set up this colony, for the sake of all the people who will live there and be happy? Or do you think the spaceship blips out of existence before it gets there?"
If you're like me, you'll find it intuitive to believe the spaceship and its colonists continue existing, even after they reach the point of no return, because claiming they cease to exist requires extra laws that wink them out of existence when they get too far away. Parsimony demands that we accept implied invisibles.
A Personal Note
Although I only just recently was convinced of this stance by Yudkowsky, the basic argument is quite old, and I really should have run into it before now. I guess the main reason I never did is because quantum physics was something I spent a lot of time on while I was still young, and I just haven't kept up with the field since I was a teenager. Since I didn't encounter Bayesian probabilities until studying philosophy of science at Spring Hill College, I just never put two and two together. Let this be a lesson to myself to never stop learning, and always update old beliefs with newly acquired information. (c:
Oh, and by the way, if you're wondering why I called this write-up succinct, you should see how much Yudkowsky writes on the topic.