OK, I found a similar problem in another thread. My error was setting up Fetchmail to use the Zimbra user as the user for the fetching process.
For those who know as much as I did or less
Fetchmail requires a local user for the rights under which to run its process. That user has to have a .fetchmailrc file in their home directory, which is where the configuration settings get saved. This user doesn't have to have anything at all to do with your Zimbra mail address to which the fetched mail will be sent.
Anyway, using the user "zimbra" as that user appears to work when you're setting up the fetchmail process, but apparently the rights are such that you can't write the mail to that directory. Using any other local Linux user does the trick.