Mail - Where are all these Incoming Messages going?

You can't view them in Mail.app or you want to look for them on the computer?

The Mailboxes are saved under your User folder, specifically your user library folder.

I don't recommend going there to look at any of the files, because one wrong click or key combo, and you have to redo your entire email setup and potentially lose all of your emails (that's if you don't have them saved on the server).

Do you have webmail access to that same account? Using mobileme mail, I see the same thing when the Mail client is synchronising from the server after I've been on the web interface all day, which includes all mails in the Sent and Trash folders. So it's "incoming" mail rather than new, received mail.

If you connect via IMAP or HTTP(S) it'll happen often. Nothing to worry about. It's because it syncs often. POP only pulls down at intervals you tell it too, or when you hit Get Mail. The other protocols don't do that...

I checked MAIL, and didn't see where you could turn off the display of the syncs; meaning, somehow suppress the notification of the sync running. I'm sure there's a command line hack, etc. for it. I didn't Google it, but can try...

You could hide the Mail Activity panel, but then you wouldn't see new mail arriving. You get the mail icon update though. I stopped noticing the activity after a while, so I just leave it displayed.