The only way to import mesh data is with the ScanTo3D tool, found with SolidWorks Premium. VRML is mesh data, not NURBS data, and so it is not native CAD data. ScanTo3D imports the mesh cloud, and has tools to easily re-create the surfaces into SolidWorks native data.
Your part is being imported as a Graphics Body.
It is a virtual image of the part.
Change "Files of Type" to VRML (*.wrl)
Change "Import As" to "Surface Body"
Open the file.
Go get a cup of coffee while it imports.
When it finishes, do NOT run "Import Diagnostics".
Save the part.
If you want a solid part, right-click on Surface-Imported1 and select "Import Diagnostics".
Be prepared for it to take hours (and hours).