ummm makes a really good point about out emails being the new poison pen letter.
Police used to investigate threats made by poison pen and they need to react to threatening emails in much the same way - and in this case, there was evidence that the threat was 'credible'.
By this, I am assuming that the writer provided information that the general public wouldn't be privy to. For example, "Be careful when you get into your car, with the registration XXX XXXXX, and your wife Celia and her sister might want to change their plans to travel to Edinburgh this weekend, because etc etc".
This would make the threat credible, because they contain information that isn't commonly known.
This should be fairly easy to investigate. The email will contain metadata which includes the IP address of the computer it was sent from. From there, the police can identify the location of the sender.