I think a big part of it is that I don't really care about other people except instrumentally. I care terminally about myself, but only because I experience my own thoughts and feelings first-hand. If I knew I were going to be branched, then I'd care about both copies in advance as both are valid continuations of my current sensory stream. However, once the branch had taken place, both copies would immediately stop caring about the other (although I expect they would still practice altruistic behavior towards each other for decision-theoretic reasons). I suspect this has also influenced my sense of morality: I've never been attracted to total utilitarianism, as I've never been able to see why the existence of X people should be considered superior to the existence of Y < X equally satisfied people.

So yeah, that's part of it, but not all of it (if that were the extent of it, I'd be indifferent to the existence of copies, not opposed to it). The rest is hard to put into words, and I suspect that even were I to succeed in doing so I'd only have succeeded in manufacturing a verbal rationalization. Part of it is instrumental, each copy would be a potential competitor, but that's insufficient to explain my feelings on the matter. This wouldn't be applicable to, say, the Many-Worlds Interpretation of quantum mechanics, and yet I'm still bothered by that interpretation as it implies constant branching of my identity. So in the end, I think that I can't offer a verbal justification for this preference precisely because it's a terminal preference.