Right-click with a file selected and you will see the available options to edit/remove it or to add new assets. Right-click with nothing selected to just add assets. The bottom right section of the asset browser also has a “+” button from which you can add assets.
Move files by dragging and dropping them to another folder or right-click and select “cut” and then “paste” it in the desired location.
Files can also be moved in the OS browser as well, but make sure to click the update button in the bottom right corner of the browser to update it.
Navigating the Asset Browser.