It is really important for those that open carry to know if they live in Stop and Identify states and the state laws about it.
http://en.wikipedia.org/wiki/Stop_and_Identify_statutes (The Detention section is the most pertinent since the cop has detained you because of your gun)
Missouri is listed as a stop and identify state so a cop asking for your ID without direct suspicion of a crime is generally acceptable under state law. The other videos that you see where the open carrier refuses to show ID are usually in states without stop and identify laws and therefore the cop has no legal authority to check your ID solely on basis of open carrying (if open carry is legal in that state).
The reasons for the ID checks are to run the person through NCIC to check for felony convictions or other notations that would make possession of the gun a criminal offense, thus leading to seizure of the gun and arrest of the carrier. Of course it's also a check for warrants and other "Wanted" notations unrelated to the open carry.
In my state, there is no stop and ID law and open carry is 100% legal so I would handle that same situation differently than someone in Missouri.