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?
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.
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.”
Phrases like a axe, a ear of corn, and a uncle occur frequently in the CSME. Examples like a address (with a following unstressed syllable) are sporadic as well.
I found more information in the article Grammar and Syntax of Smoky Mountain English.
This is just one dialect; I suspect it occurs in many others but likely less frequently. Though I live in the north (and therefore speak a significantly different American dialect), I also notice myself using "a" before vowel sounds sometimes.
Sorry, but that's poppycock. Can you cite any research article that backs that up?
El-Koumy, A. S. A. (1998). Effect of dialogue journal writing on EFL students’ speaking skill. Retrieved from ERIC database (ED424772)
Rokni, S. J. A., & Seifi, A. (2014). Dialog journal writing and its effect on learners’ speaking accuracy and fluency. Study in English language teaching, 2(1), 28-37.
Kim, Y. (2008). The effects of integrated language-based instruction in elementary ESL learning. Modern Language Journal, 92, 432-451.
Nation, I. S. P., & Newton, J. (2009). Teaching ESL/EFL listening and speaking. New York, NY: Routledge. 
Newton, J. (1995). Task-based interaction and incidental vocabulary learning: A case study. Second Language Research, 11(2), 159-176
Blake, C. (2009). Potential of text-based internet chats for improving oral fluency in a second language. Modern Language Journal, 93, 227-240.
Sanchez, M. A. (2014). The effect of written input on young EFL learners’ oral output. Journal of English studies, 12, 7-33. DOI:10.18172/jes.2821
Fathali, S., & Sotoudehnama, E. (2015). The impact of guided writing practice on the speaking proficiency and attitude of EFL elementary learners. The journal of teaching language skills, 7(1), 1-25. DOI: https://dx.doi.org/10.22099/jtls.2015.3456
“. . . almost all the above previous studies concluded that writing abilities affect speaking proficiency in a positive way (e.g., El-koumy,1998; Kim, 2008; Williams, 2008; Blake, 2009; Nation & Newton, 2009; Rokni & Seifi, 2014; Sanchez, 2014; Fathali & Sotoudehnama, 2015). Journal writing, integrated based instruction of speaking and writing, written input and output, and guided writing practice are likely to have a positive impact on learners’ speaking abilities. Hence, ‘writing to learn’ is proven by evidence from the previous studies to enhance spoken language.”
— Speaking and Writing Interconnections: A Systematic Review Fouad Akki, Mohammed Larouz Department of English Studies, School of Arts and Humanities, Moulay Ismail University, Morocco https://pdfs.semanticscholar.org/3aad/04eb4c1f0fa690abc1aea4033d22cb961c4e.pdf
I have my own test:
A student who is taught to write, “An apple a day keeps the doctor away. Yesterday, I gave my teacher an apple, and today, my teacher gave me an apple” is highly unlikely to say even in casual speech, “A apple a day keeps the doctor away. Yesterday, I gave my teacher a apple, and yesterday, my teacher gave me a apple.” Ergo, correct writing influences correct speaking.
Of course, the exception might be the 60% of college freshman who have to enroll in remedial English courses, 95% of whom never go on to complete their degree. I don’t blame the students; I blame their teachers. And beyond the teachers, I blame those who might have influenced educational policy, such as those claiming, “Literacy means being to able to read, not speak.”
We reap what we sow.