Are you using POP, or IMAP?
If it's IMAP, seems to me that actual message "content" isn't downloaded to your drive, just the "message headers" (which you see in the Mail "inbox").
When you click a header, it downloads the content so you can read it, but I don't believe it actually saves "a copy" of the message to your [local] drive. In other words, when you quit Mail, the downloaded message will be gone (although the msg header will still be there).
To save the content of a message to your drive, you need to create a mailbox in the "on my Mac" area. Then, drag the message from the inbox to the mailbox you created, and the content WILL be downloaded and stored locally.
I may have this wrong, and I welcome correction from others who may explain the process better.