To upload a file, you can simply drag-and-drop it into the browser window. It will automatically be uploaded to your home directory.
Alternatively, pull up the side-menu/clipboard-pane [ Shift + Control +Alt(Opt) ] and double-click on the drive under "Devices". You can then navigate the filesystem by double-clicking through directories and clicking the "Upload Files" button.
Note: File upload can be pretty slow so it is recommended that you use a different method for files larger than 100 MB.