The word "authentic" is tricky. It means "real" but is used in certain situations. You use "authentic" to talk about international food, for example:

Do you know of any good, authentic Vietnamese restaurants here in the city?

I also use the word "authentic" to describe the kind of English sentences I try to write on this website:

On PhraseMix.com, I try to teach authentic English that you can use in the real world.

So an "authentic experience" is an experience that seems real and natural for that situation. People who want an "authentic experience" in a foreign country want to eat at the same restaurants that locals eat at, talk to locals who aren't paid to be tour guides, and see what life is like in that location.