I have noticed that a lot of native American speakers use the indefinite article "a" in front of words beginning with vowels, such as interesting, old, apple , etc. Is there any reasonable explanation?
There are many different accents in North American English. In some of those accents, it might be quite comfortable for the speaker to say “a apple.” Either because of the way they are putting their syllables together, or perhaps because they are from a region where people tend to speak very slowly. It might be “a [pause] apple.”
I use a with such words and I am a native speaker of American English. The only sensible explanation is that language usage changes and that it sounds fine to me.
It used to be that people would use an in contexts where we, or not all native speakers (still) do, such as before some words starting with h- (an hundred) or other sounds, such as the yu- in eunuch. I don't talk much about eunuchs but when I do I say a eunuch, not an eunuch as some people, such as Shakespeare, does in Twelfth Night, performed first in 1600, give or take a year.
There's a theory that says there's a gottal stop between a and apple (etc) which "licenses" such usage. But I'm not self aware enough to determine if I employ one or not. I certainly don't say a educated man like I would Hawai'i with its prominent gottal stop.