Does the user have to do anything after receiving the invitation?

Yes, the user must accept your invitation to activate the email address. Until the user does this, messages to this email address will be returned to sender.

When will the email address be active?

The email address will be active as soon as the user accepts your invitation. As soon as this is done, the user can begin sending and receiving email. To check the user's activation status, visit the Email Control Panel. If the status is pending, make sure the person received your invitation (did you send it to the correct address?), and if it did arrive, remind the user to activate it.