If you are set up to do a single-file download in your form, then you can fill in section 9.1 of the EBD Settings if you want to do a custom message in the email rather than the default message. In that section, you can use the [file_name] tag.
If you are set up to do multiple-file download (as in your are using the <ebd/> or <ebd_left /> tags in order to get radio buttons or checkboxes), then you can fill in section 9.2 of the EBD Settings if you want to do a custom message in the email rather than the default message. In that section, you can use the [file_urls] tag, which will list all the file names selected by the user and make them links that he can click on to download.
Note that both of those sections are for the email that gets sent to the user. They don’t affect the display of the links on the webpage itself (if you selected “both” or “inline”).