How to Transfer Media Files with Media CategoriesEnhanced Media Library

To export/import the content itself (including categories) WordPress has inbuilt tools, see admin menu Tools > Export and Tools > Import.

To export/import your website content and EML settings you should do both: export/import the content in Tools menu, and then export/import EML settings in Settings > EML Utilities menu.

If you wish to move media categories and media files only, you have to use WordPress inbuilt export/import and choose Media instead of All content at Choose what to export.