According to Wikipedia the association of Occam's razor to many worlds goes back to Hugh Everett who originated the idea of many worlds in 1957:
Since the wavefunction merely appears to have collapsed then, Everett reasoned, there was no need to actually assume that it had collapsed. And so, invoking Occam's razor, he removed the postulate of wavefunction collapse from the theory.
According to Wikipedia those who reject many worlds also use Occam's razor:
Occam's razor rules against a plethora of unobservable universes – Occam would prefer just one universe; i.e., any non-MWI.
And those supporting many worlds claim in response to this objection:
MWI response: Occam's razor actually is a constraint on the complexity of physical theory, not on the number of universes. MWI is a simpler theory since it has fewer postulates. Occams's razor is often cited by MWI adherents as an advantage of MWI.
If one wants to use Occam's razor to get a theory with the least amount of complexity, one could always use the philosophical theory of Occasionalism. Wikipedia describes this explanation of why things are the way they are as follows:
Occasionalism is a philosophical theory about causation which says that created substances cannot be efficient causes of events. Instead, all events are taken to be caused directly by God.
The theory with the least complexity, if that is what is most important, could be paraphrased as "God did it".
Wikipedia contributors. (2019, July 22). Many-worlds interpretation. In Wikipedia, The Free Encyclopedia. Retrieved 13:11, July 24, 2019, from https://en.wikipedia.org/w/index.php?title=Many-worlds_interpretation&oldid=907411360
Wikipedia contributors. (2019, June 10). Occasionalism. In Wikipedia, The Free Encyclopedia. Retrieved 13:19, July 24, 2019, from https://en.wikipedia.org/w/index.php?title=Occasionalism&oldid=901274339