I can't read in a SolidWorks *.sldasm file by itself, but eDrawings can, why is that?

A SolidWorks *.sldasm assembly file does not contain any "true" geometry. It simply refers to *.sldprt files - which are the individual part files that do contain the geometry.  As TransMagic is primarily a 3D geometric translation and Repair application, it needs the *.sldprt files that contain the actual CAD geometry to be present when reading a *.sldasm file.
One confusing aspect of a *.sldasm file is that it does contain a "visual" representation of the assembly that you can load into eDrawings and other viewers; however, rest assured none of the CAD geometry is in the *.sldasm file.