Sorry I can't think of any more specific examples. I can't even remember where I've heard this beyond reading in books. Hoping someone can add some context to this for me.
The word choice that you mention is often known as the "Royal We" and historically as "pluralis maiestatis".
It has the implication that whatever the speaker says is not just a personal statement, but as a monarch it becomes the collective opinion of the state as well. In some sense it imposes itself onto the listener as well as the speaker.