Hello, and welcome to the Discussions.
It is AOL's policy to remove messages from your Inbox, which since their accounts are IMAP type, is on the server, after 27 days. There is no action of Mail to prevent this. AOL does provide another folder on the server named Saved -- you will see it in the Sidebar further down, into which you can move messages prior to 27 days. Or you can move messages to an On My Mac mailbox you can create as a New Mailbox, where they will be saved on that one computer only.
Ernie