In the large islands south-east of the Asian mainland there’s a better explanation than most. They have komodo dragons, enormous monitor lizards that can eat a whole pig. Admittedly, they have no wings (although dragons don’t necessarily have to have them) and don’t breathe fire, but they are massive, dangerous reptiles. The imaginative leap from a komodo dragon to a real dragon is much smaller than in, say, England.
England is a green (and wet) and pleasant land, but it most certainly is not troubled by marauding reptiles. We have almost no snakes, and of those we do have only one, the adder, is properly dangerous. However, the explanation could be the Book of Revelation (easily my favourite bit of the Bible, due to its bonkers and entertaining imagery). In Revelation, Satan turns into a huge dragon with… 7 heads and 13 horns, I think.
That does beg the question of where the Biblical dragon came from, though. I think the Zoroastrians wrote of Azhi Dahaki, the splendidly named super-dragon that was the offspring of the Father of Lies. The bad news is that at the end of the world Azhi Dahaki breaks free of his bonds and destroys the world.
Which sounds a bit like Revelation. Norse mythology also features dragons, specifically Nidhogg, who chews on the roots of the world-tree, Yggdrasill.
The Far East has its dragons too. The Chinese dragon is pretty famous, and participated in the race to decide the order of the Chinese zodiac. (The dragon came fifth, incidentally, after helping people out by creating rain and assisting a rabbit [who came fourth] reach safety).
Religion seems the common thread between the differing cultures, a common source for stories involving dragons.
Of course, dragons have been popular for a long time, from the Zoroastrians to Christianity and Beowulf, all the way to the present day.
They’ve long been a staple of fantasy as well. The Lord of the Rings does not feature any dragon prominently (I think the Nazgul ride them later on), but they have a larger role in the Silmarillion. More recently, George RR Martin makes excellent use of them in A Song of Ice and Fire (review probably coming on the 21st) and Skyrim will feature them heavily.
Why do they remain so popular?
Probably because a massive, flying, evil, fire-breathing reptile is cool. Just as fantasy has dwarves (likely grumpy smiths with beards) and elves (slender, snobbish archers) it needs a big bad wolf. And that’s the dragon.