Many worlds interpretation: adding approximately zero explanatory value, at the low low cost of infinite parallel universes.
I think the best way to understand MWI is that a lot of physicists/philosophers are more comfortable with the idea of inhabiting an infinity of forking but "real" (essentially classical) universes than they are with inhabiting a single "unreal" universe in which QM is taken seriously.
That's a pretty uncharitable interpretation of it. If you look at the actual physics, MWI is incredibly simple: the wavefunction just evolves as the wavefunction evolves. That's it. There's some amplitude over here, and there's some amplitude over there, and this possibility and that possibility are mutually exclusive etc (with some "weird" quantum correlations, of course.)
The Copenhagen interpretation's "wavefunction collapse" is just like a Bayesian update -- the wavefunction you're left with is like a conditional probability. Maybe it makes you happy to throw away the "other universe" bits of the wavefunction and normalise probabilities on your own branch, but you obviously don't have to, and any discussion of when the collapse happens (or worse, whether it's a physical process that propagates across the universe faster than light) is obviously very silly business.
Any other intepretation requires new, undiscovered physics. You have to make additional assumptions in order for the equations of quantum mechanics not to give rise to many world.
I think the best way to understand MWI is that a lot of physicists/philosophers are more comfortable with the idea of inhabiting an infinity of forking but "real" (essentially classical) universes than they are with inhabiting a single "unreal" universe in which QM is taken seriously.