There has never been an official statement of why the companies were switched.
In 2015, it was announced that Shout! Factory had purchased all rights to Mystery Science Theater 3000 from Best Brains, Inc. for an undisclosed sum.
Shout! Factory releases
Shout's rights to MST3K coincided with the 20th anniversary of the original airing of the show; therefore, their first release was a spectacular celebration of the show. They have since resumed normal numbered releases, continuing their numbering where Rhino left off. Notably, Shout uses Roman numerals to number their sets. They have also released several previously-available episodes on individual DVDs.
See also: List of Shout! Factory DVDs