Induction. You have uncertainty about the extent to which you care about different universes. If it turns out you don't care about the born rule for one reason or another the universe you observe is an absurdly (as in probably-a-Boltzmann-brain absurd) tiny sliver of the multiverse, but if you do, it's still an absurdly tiny sliver but immensely less so. You should anticipate as if the born rule is true, because if you don't almost only care about world where it is true, then you care almost nothing about the current world, and being wrong in it doesn't matter, relatively to otherwise.

Hmm, I'm terrible at explaining this stuff. But the tl;dr is basically that there's this long complicated reason why you should anticipate and act this way and thus it's true in the "the simple truth" sense, that's mostly tangential to if it's "true" in some specific philosophy paper sense.