"봐" literally means "look" but in a sentence, it makes sentence less strictly. If you omit "봐" in the sentence "말해봐" as "말해", it's like an order not advice.

as Goo Young said '봐' is from the verb '보다' which means 'to look'. When you say '말해봐' it is '말하다' + '보다' (speak and look). You are suggesting to someone to give it a try like in those sentences : '먹어봐' : eat and see / eat and look / try to eat '나가봐' : go out and see / go out and look / try to go out and so on... i think you are making a mistake, it is '보세요' not '부세요' '보세요' is polite form of '봐' you use '나가봐' with friends and relatives you use '나가보세요' with new relations and older people