The organisation itself needs an article. 'BBC' isn't a name like 'italki' or 'Dior' or 'Mitsubishi' . It's the British Broadcasting Corporation. There are many corporations in the world, but only one British Broadcasting Corporation - hence the article. When 'BBC' is used before another noun, either as an adjective or in a compound noun, you don't always need the article - for example 'I met some BBC reporters.' It's equivalent to 'the US' (meaning the United States), but 'US dollar'. When an entity that needs an article is used adjectivally or as a compound noun, you can lose the article.

You should always say "the BBC". The first sentence is wrong. The reason why it's called '"the BBC" is simply that it has been named "the BBC". "Italki" isn't called "the Italki" because the creators of Italki chose not to call it "the Italki" (and for good reason: it sounds bizarre, since "italki" isn't a word that means anything). "Facebook" was originally called "the Facebook", but the creators later changed the name to just "Facebook". As a general rule, a proper noun is more likely to include "the" if the name actually means something ("the Italki" sounds bizarre because you can't have "an italki"), but whether "the" is part of a name is completely up to the person making the name. "BBC" stands for "British Broadcasting Company", so it sounds natural to add "the" since you can just as easily say "the/an British broadcasting company" (ie. the name actually means something). The BBC could have also chosen to just call themselves "BBC" (without "the"), but they didn't. The most important thing to remember is that "the" is never written *before* a proper noun, but it may be *part* of the proper noun the same way the two Bs and the C are part of the thing called "the BBC".