I can't really speak for them, but I somewhat believe that they believe what I believe, i.e., that the many-worlds interpretation of nonrelativistic QM models the world as a function from a (usually nondiscrete) configuration space to the complex numbers, and the norm-squared of that function is the measure that predicts the outcomes of experiments. So there's no well-defined way to count worlds in general, although in some circumstances it may be helpful to refer to a small region of the configuration space as a "world", and different "worlds" can have different measures.
From the Quantum Physics Sequence:
Decoherence is implicit in quantum physics, not an extra postulate on top of it, and quantum physics is continuous. Thus, "decoherence" is not an all-or-nothing phenomenon—there's no sharp cutoff point. Given two blobs, there's a quantitative amount of amplitude that can flow into identical configurations between them. This quantum interference diminishes down to an exponentially tiny infinitesimal as the two blobs separate in configuration space.
I'm still confused by the Born rule. (The worlds where the Born rule makes good predictions have a lot of L^2 measure. But they have very little L^1 measure! The lion's share of the L^1 measure is held by worlds where an L^1 version of the Born rule holds. Why does our experience accord with the Born rule?) But I have more reading to do.
I don't know much about Mangled Worlds.
What privileges L^1 measure if not world-counting? I just looked up mangled worlds again, and Hanson explicitly uses world-counting; at least, he seems to admit that worlds don't have well-defined numbers, but he thinks this isn't a problem for reasons I don't really understand.
If you believe the MWI [1] you should care about the future a lot more than the present. Imagine you're considering whether to take a break and eat some chocolate in an hour or in two. You'll get similar enjoyment out of both choices, so you might think it doesn't matter. But if every quantum event between one and two hours from now will branch the universe, and there are lots of such events, in two hours there would be hugely many more yous to experience your chocolate break than in only one hour. The MWI implies we should be willing to make substantial sacrifices in terms of current happiness for the benefit of our future selves. In other words, your preference for investing probably isn't strong enough.
In trying to apply this to altruism you do need to be careful. Some charities are more like spending, in that their benefits are mostly in the present, while others are like investing. If I donate to the Against Malaria Foundation to distribute mosquito nets, the main benefits are preventing current or near-future people from dying. There are probably some long term effects, like a stronger economy when you have fewer people sick, but they're not the goal or the main effect. On the other hand the Future of Humanity Institute, a charity trying to prevent existential risk, is much more like an investment in that nearly all its benefit (which is really hard to predict or quantify) goes to future people. Metacharities promoting effective altruism, like 80,000 hours, Giving What We Can, and GiveWell, are another sort of investment-like charity, influencing people's future giving. And then there's the option of straight up monetary investing now and donating later.
If you accept the MWI you should be evaluating your altruistic options primarily on their future effects, with more emphasis on farther-future ones.
I also posted this on my blog
[1] Which I still don't know enough about to have an opinion on the truth of.