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> there are different branches

If by different branches you mean something like « now there is one, now there are two » it seems that branching is no less mysterious than collapse.

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Branching comes from decoherence where quantum entanglements spread to the environment so the coherence of the entangled environment is suppressed and we only observe what looks like a classical world unless we isolate systems to see their coherence. MWI is just saying that entanglement does not stop at measurement, and that decoherence spreads it to the universe.

So there are "different branches" as in "one branch now, two different branches later" or not? (Or one gazillion branches now, one gazillion branches plus one later. The point is whether the "different branches" can be counted or not.)

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