Hi. I'm Brian and I will do my best to help you.
The option to keep messages on the server is available only for POP accounts. The default behavior of the POP protocol ia do download messages from Inbox and then delete them. The "leave messages on the server" option prevents that deletion.
With AOL, the usual account type is IMAP. The IMAP protocol always leaves messages on the server, but it's a synching protocol so that if you deliberately delete a message in the client (Outlook) you'll also delete it from the server.