In layman's terms (to the best of my understanding), the proof is:
Copenhagen interpretation is "there is wave propagation and then collapse" and thus requires a description of how collapse happens. MWI is "there is wave propagation", and thus has fewer rules, and thus is simpler (in that sense).
and thus requires a description of how collapse happens
... which it doesn't provide.
If you know more Eliezer Yudkowsky facts, post them in the comments.