For one thing it allows one to log in under any email you own; for example on Gravatar I can log in on both my primary X@gmail.com and secondary Y@gmail.com addresses, and both log "me" in because both are claimed by me and associated.
This makes particular sense with email related clients; Gravatar's settings are dependent on your email, so it makes perfect sense to add multiple emails. For other clients, even if you can't use more than one email within the app, they serve as extra logins so you don't forget "Oh, which email did I sign up for Facebook with?"
When an app lets you set a "primary" email, generally that means that's the email which will get all notifications/newsletters/ect from the service. Setting up multiple emails can be used to easily change this in case you deactivate or rarely use an email.