What does a broken hyoid bone indicate about the cause of death

The answer is strangulation 

Usually a broken hyoid, because of its role in keeping the airway from caving in and in the most nominal in degree protecting the airway, the cause of death usually indicated with a broken hyoid is indicative of some form of strangulation usually with some form of high energy usually via hanging.

Mostly people with this type of injury may also be internally decapitated in which case the die from Decapitation. Or they die from the inability to bring oxygen into the body or a process known ad asphyxiation.