It doesn't explain it. Neither does any interpretation of quantum mechanics that reduces quantum amplitudes to classical probabilities.
Suppose I claim that a coin comes up heads with some classical probability $p\in(0,1)$. What does that mean, operationally? In other words, how can you test it? You can flip the coin $n$ times, and if I'm right you should get $np$ heads. Actually, that's not true, you could get any number of heads. Well, at least certain numbers of heads are more probable than others, assuming my claim is true. You can prove that if my claim is true then if you flip the coin $n$ times then you'll get between $k$ and $\ell$ heads with probability $q$, where $q$ is a complicated function of $p$, $n$, $k$ and $\ell$. The trouble is that this new probabilistic claim is on exactly the same footing as the original one, and if you try to give an operational meaning to it, you get an infinite regress.
In practice, we deal with this problem by inventing a cutoff $ε>0$ and reasoning as though anything with a probability less than $ε$ is certain to not happen. But there's no way to formalize this idea without destroying the internal consistency of the probability calculus.
The problem in many-worlds is exactly the same, and can be "solved" in exactly the same way, by inventing an $ε>0$ and deciding that worlds with amplitudes within $ε$ of $0$ don't really exist. This isn't a solution at all, but neither is adding the Born rule to the theory. That merely substitutes the classical-probabilistic version of the problem for the quantum-probabilistic version.
Deterministic theories like Bohmian mechanics have essentially the same problem. Among all the possible worlds (given by different initial conditions), there's no obvious reason why the actual world should be one in which experiments point us to the correct laws of nature.