Well the answer to that question actually depends where in the world you are dining.
The Royal Family are correctly eating in a style named Continental European. Table manners and dining etiquette actually vary between countries so placing hands on a dining table can be considered either rude or essential.
This particular style of dining etiquette dictates that hands and wrists should be placed on the dining table while eating, remaining visible at all times. It is frequently used in France, Spain and throughout most of Europe. Here's why...
As with most etiquette guidelines, particularly table manners, there is a historical trail behind the rule and in this case, it was a political reason. In the XVII Century, Louis XIV discovered a conspiracy to poison him with arsenic. Concerned that one of his guests at the dining table would try to kill him, he ordered that everyone keep their hands visible and placed on the table throughout every meal. As people wanted to imitate the aristocracy at that time, the rest of the country soon followed suit.
Don't expect to see much of this happening at a British dining table though, the custom spread throughout Europe but never reached Britain. Hands are always be placed on the lap when not eating.
The Etiquette Consultant