I don't think that the 'rule' specifically applies to the (so-called) Royal family. It's just that they tend to observe traditional (often more formal) ways of doing things than many other people do. That can apply as much as to the social protocols of when (and when not) a hat should be worn, or the way that the Port should be passed at the table, as it can to things like funerals.
It used to be considered extremely 'un-British' to display any strong emotion (particularly sorrow) in public . The rule was always to 'remember to keep a stiff upper lip at all times'. That was taught in all schools (but particularly in public schools) and anyone seen crying at a funeral would have been seen as 'letting the side down'.
The rest of the country seems to have moved to a more 'continental' position but many Royalists probably still expect the Royal family to be as 'traditional' as possible. Indeed, for many,the whole point of the Royal family is that they represent 'tradition', so they can't be seen to deviate from it. (The Queen would probably rather be somewhere sunny at Christmas, than stuck at Sandringham every year, but tradition dictates that she has to go to Norfolk, whether she likes it or not!).
Chris