I don't think that's true - it takes that into account, it just looks like it collapses because you're in the world with that state - not seeing the entire picture. MWI accounts for it appearing that way while in actuality the wave function never collapses.

Max Tegmark gives a pretty accessible description of this in his book - Our Mathematical Universe, and Eliezer Yudkowsky does a decent job with a general explanation here (though he avoids talking about MWI): http://lesswrong.com/lw/pd/configurations_and_amplitude/

I think the Copenhagen interpretation is now considered wrong by most physicists. It's strange to me that it was ever really thought of as true - though it makes sense that it would look like what's happening at first.

It's less that MWI is a predictive theory and more that MWI is something that comes out as a result of the theory's prediction (Schrodinger's wave equation that doesn't collapse).