This sentence is a little bit long, but we can break it into smaller chunks. Each piece is much easier to understand.
Below is a delightful kinetic typography video made by a clever graphic artist called Matthew Rogers. Please notice that the movie includes closed captions in Russian (just click on CC to turn them on or off):
In this video, Stephen Fry is talking about "correct" or "incorrect" use of language. Many teachers and "self-appointed opinion leaders" (= people who spend too much time reading dusty old grammar books and out-of-date dictionaries) consider themselves to be superior because they like to use old-fashioned super-correct grammar and spelling.
These annoying people enjoy sneering at other people's "mistakes" and trying to correct them but Mr. Fry points out that these silly people are often quite ignorant. The so-called "mistakes" these pompous fools worry about are evidence of language change, something that arrogant opinion leaders refuse to accept.
Mr. Fry goes on to explain that there are rules for how to speak or write in different situations (the technical word for this is pragmatics). However, these are social rules, not grammatical rules. In other words, "correct" grammar, spelling and vocabulary depends on who you are speaking to.
self-appointed = they choose themselves (if you say this about someone, it usually means that you don't think they deserve respect):
1. "Everybody was suddenly a self-appointed expert" [The speaker means that these people are not qualified to give advice]
2. "Our self-appointed tour guide took us to an expensive souvenir shop" [The speaker thinks this person is not a real tour guide; perhaps he works for the souvenir shop]
sneer at = if you sneer at someone, you use your face or your words to show that they are low class or worthless:
1. "In Paris, some people sneer at Americans who speak French poorly, but in the countryside, people are usually much more friendly." [Some over-educated French people are unhappy because the US is stronger and richer than their country. Sneering at Americans make them feel better]
2. "The waiters will sneer at your choice of wine" [In some expensive high-class restaurants, waiters will not respect you if you don't choose the correct wine to go with your food]
pompous = if you say someone is pompous, they have a very good opinion of themselves, but you think they look stupid or ridiculous.