Broadly, branches become less likely to interact as they become more dissimilar; dissimilarity tends to bring about dissimilarity, so they can quickly diverge to a point where the probability of further interaction is negligible. Note that at least as Eliezer tells it, branches aren't ontologically basic objects in MWI any more than chairs are; they're rough high-level abstractions we imagine when we think about MWI. If you want a less confused understanding than this, I don't have a better suggestion than actually reading the quantum physics sequence!

To expand on what pragmatist said: The wavefunction started off concentrated in a tiny corner of a ridiculously high-dimensional space (configuration space has several dimensions for every particle), and then spread out in a very non-uniform way as time passed. In many cases, the wavefunction's rule for spreading out (the Schrödinger equation) allows for two "blobs" to "separate" and then "collide again" (thus the two-split experiment, Feynman paths and all sorts of wavelike behavior). The quote marks around these are because it's not ever like perfect physical separation, more like the way that the pointwise sum of two Gaussian functions with very different means looks like two "separated" blobs. But certain kinds of interactions (especially those that lead to a cascade of other interactions) correspond to those blobs "losing" each other. And if they do so, then it's highly unlikely they'll accidentally "collide" again later. (A random walk in a high-dimensional space never finds its way back, heuristically speaking.) So, as long as the universe has relatively low entropy (as it will until what we would call the end of the universe), significant interference with "blobs" of wavefunction that "split off of our blob" in macroscopic ways a long time ago would be fantastically unlikely. Not impossible, just "a whale and a petunia spontaneously appear out of quantum noise" degree of improbability.