After you've authorized the transfer, you're going to get another email from GoDaddy begging you to not leave. Ignore that email and go back to GoDaddy.com one last time. Once there, go to your Domain Manager and click on "Domain Transfers." Once there, check the box next to the domain you're transferring and click "Accept," and then you're all done.
Don't fret if some of the steps in which you wait on a response from GoDaddy take several hours. They're stalling in the hopes that you'll give up. Don't give up, because you're doing the right thing.