Nope. You are right. BBC slips up. :)

Seen yes, boosted no. Explode is solidly a verb, nothing nouny about it.

"explosion", would be the noun you are thinking about. language is the noun here and they say that it is expanding, or gaining popularity.

actually, this one's tricky. there is no "explode" noun existing, nevertheless, there is an adverb "explode". BBC is ok, but i haven't seen such order of the words in a sentence for a long time, but it's ok. consider this as a "gang..style has boosted the popularity of the korean language and that has led to the fact that it has exploded". this one sounds veeery wrong, and that's why you wanna use the adverb "explode" instead.

1) A desire to learn the lyrics of K-Pop hits like Gangnam Style has boosted the Korean language's popularity in countries like the US, Canada, Thailand and Malaysia. 2)A desire to learn the lyrics of K-Pop hits like Gangnam Style has SEEN the Korean language's popularity EXPLODE in countries like the US, Canada, Thailand and Malaysia. it's just a simple case of wrong word choice where if you use sentence (1) you need to remove explode in the second part. And in sentence(2) you need to replace boosted with seen. you are confusing your self when thinking about use as a noun.