What does the "Many-Worlds Interpretation" of Quantum Mechanics even mean?

Sean Carroll identifies some Wrong Objections to the Many-Worlds Interpretation of Quantum Mechanics, which is fine but I'd rather he addressed some of the better ones.

I have always thought of (my own experience of) the universe as corresponding to (a very small part of) one particular configuration of a stochastic system, and that having a theoretical model for that system allows me to predict conditional probabilities of certain features (measurements) given others (state preparations). I suppose other configurations could be regarded as alternate worlds which *could* in some sense exist. But why is it necessary (and in fact, what would it mean) to suggest that they *do* exist?

Oh dear! Now I feel a little Feynmanesque "poem" coming on:

We don't know the meaning of "meaning",
And we don't know the meaning of "is".
So how can we possibly claim to know
What the meaning of  "the meaning of "is"" is?

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