You can list a remote FTP directory with curl by making sure the URL ends with a trailing slash. If the URL ends with a slash, curl will presume that it is a directory you want to list. If it is not actually a directory, you will most likely instead get an error.
With FTP there is no standard syntax for the directory output that is returned for this sort of command that uses the standard FTP command LIST. The listing is usually humanly readable and perfectly understandable but you will see that different servers will return the listing in slightly different ways.
One way to get just a listing of all the names in a directory and thus to avoid the special formatting of the regular directory listings is to tell curl to --list-only (or just -l). curl then issues the NLST FTP command instead:
curl --list-only ftp://ftp.example.com/directory/
NLST has its own quirks though, as some FTP servers list only actual files in their response to NLST; they do not include directories and symbolic links!