What Gives Them The Right
What gives them the right to look at other people's private emails is that those e-mails were sent to someone who was threatened by the message, and called the FBI to investigate. Those e-mails also contained non-public information about the comings and goings of U.S. Armed Forces officers.
When the FBI investigated, they found just cause to get a legal warrant to find the sender, and inspect the sender's e-mail. It was all legal, and constitutional, and had nothing at all to do with interfering in some other country's business. This was a case between U.S. citizens.