In short the problem is that there's no explanation for why you only experience a single branch, and no explanation of why the probabilities associated with those branches are given by the Born rule.

Although there are many versions of MWI, the general claim is that a single axiom (a unitarily evolving wavefunction) is sufficient to explain our observations of the universe. Without getting into what it means to "explain" something, it seems fair to demand that a sufficiently intelligent agent with no prior knowledge of our physics should be able to predict what the theory says about their future observations. However no one would be able to make any experimental predictions based on the above axiom. There's nothing there to suggest that if the agent becomes entangled with a quantum system then they would only experience a single branch of the entanglement. Even if you add that in, there's nothing to suggest which branch the agent will experience (this is roughly the preferred basis problem). Even if you add that in, there's nothing to suggest with what probability the agent will experience that branch.

You could always add in the above as additional axioms. You could postulate some physical content to the Born-rule, or to something weaker from which to deduce the Born rule. But this would break the illusion that MWI requires fewer axioms. Instead, a great deal of energy has been expended into deducing the Born rule purely from that single axiom. There have been many very clever attempts at this, but none convincing enough to settle the matter.