I haven't seen any proof (stronger than "it seems like it") that MWI is strictly simpler to describe. One good reason to prefer it is that it is nice and continuous, and all our other scientific theories are nice and continuous - sort of a meta-science argument.
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).
If you know more Eliezer Yudkowsky facts, post them in the comments.