You are looking to integrate your MTA's (mail transfer agent = mail server's) virtual user table with something that can be managed through code running through Drupal. There are multiple ways to do that and the exact mechanics will depend heavily on the particular MTA you are using, but the general idea will be the same. As for specific instructions for your particular MTA, I'd try Google or ServerFault.
That said, if I can be allowed to use Postfix as an example, what you are looking for seems to be configured primarily through the
virtual_mailbox_maps configuration setting. Official documentation, Postfix and MySQL howto, quick guide to get you started and Virtual mailhosting guide with Postfix on Gentoo.
Once the MTA looks to a database table for address translation, adding a forwarding address becomes a simple
INSERT SQL query with appropriate values. If everything is set up correctly it should then take the e-mail that was received and redirect it to the user's real e-mail address.
Oh, and one more thing; make very sure you properly validate the user's provided e-mail address, so you don't become a source of spam for others. Send them a short e-mail with either a link to click on or a code that needs to be entered to activate the forwarder.