In addition to Peaches's answer: I think it has something to do with being tactfulness in using the language. If your friend wants to tell you something but he's hesitating for some reason, you would say '말해봐' instead of '말해'. A detective would say 말해봐 to win the accused co-operation more readily when they have no decisive evidence. If they say '말해' which is an imperative, they're forcing the accused to say the truth. '말해봐' sounds less pushy, with this one they're being someone like a good friend. We also say "한 번 봐봐", be aware that each 봐 is different in meaning. It's a milder way of saying "Have a look at it". In other cases it has other usages, but I'll hold it back till you come up with a question about it.

I'll give you a temporary answer before someone qualified gives you a better one. 보다 = to see, look, watch.....So it is useful in it's own right but it is so common because of it second usage. It is used to modify almost any action verb to mean "to try" that verb A good way to think about it is "try ____ and SEE what happens" the structure is (verb) + 아/어/여 보다 하다 -> 해 봐요. = try to do it past -> 해 봤어요. = I, he, she etc. tried to do it. Did you try it? etc. 먹다 -> 먹어 봐. = try eating/tasting this. This is so common that there are some fixed expressions and the meaning extends beyond our use of to try. 주다 -> 줘봐 give it to me, let me see it. 말하다 -> 말해 봐요. I don't know the the subtleties of this. I think it's just "tell him/her" 가봤어 = "Have you ever gone there?" "I have been there" 안 가봤어 = "I have never been there." this is really common and useful There's a lot more really common usages. I'll let some one who actually knows tell you.

I already know that it carries the meaning of "to try" in many cases. The question is about the many cases where it does not seem to be necessary to use it. What's the nuance of 말해봐 vs just saying 말해. In which situation would you say it instead of 말해? Native koreans... 말해봐. =P