Difference in [ə] pronunciation at the end of a word in British and American English

• "What causes that sound?" - one could just as well ask why the American English speakers -don't- pronounce it. A reasonable response would be that it's not spelled that way, but to counter that, there's all sorts of pronunciations in English that "aren't spelled that way".

But to answer directly, the standard dialect of British English is non-rhotic meaning that they don't pronounce 'r's at the ends of words. Except interestingly you've found where they do pronounce them in places not expected. Since a word-final 'r' is preceded by a vowel, non-rhotic speakers will tend to pronounce vowel final words with a coloring of an 'r' because that's what the 'r' final words sound like (a vowel with 'r'-coloring).

Relatedly, a non-rhotic pronunciation almost always retains some vestige of the 'r', and it comes back in some circumstances, especially if the following word starts with a vowel. The classic Boston accent, parodied by 'I parked the car in Harvard yard', pronounces 'correctly' (see below for explanation) it as

'I pAHked the cAH rin HAHvAHd yAHd'

where the 'r' of 'car' -is- pronounced -in this context (but not usually). This is the way the accent says it. If you, as a non-Bostoner, say it without the 'r' after car, you won't be saying it like someone with a Boston accent does in real life.

• "Which way is the correct way to pronounce it?" - 'correct' is what people normally use but it's just not appropriate here It's really "What is correct -in the particular language community-?" or even better "Which way do they almost always do it?". People follow rules, but they're not necessarily he ones you do, and which set is considered correct is more about sociological differences.

'What causes that sound?' A different accent. 'Which way is the correct way to pronounce it?' There isn't one.

I'm not certain, but you may be hearing Intrusive R, where R sounds are inserted at the ends of words ending in certain vowels if the following word also starts with a vowel.

Its "correctness" in the context of non-rhotic accents is debatable, but it is certainly not how anyone with a rhotic accent speaks, such as someone from San Diego.