[Greg Marra]

Ah, you have to tell Solidworks what to do very explicitly. When opening the file, switch the filter to STLs, then click the file, then an "options" button appears. Choose to import as a Solid Body instead of as a Graphics Body. Then you can interact with it in Solidworks. This will limit you to a 20,000 face STL, but that's more than enough to make my old laptop cry.