Why is "ladies and gentlemen" translated as "meine Damen und Herren"?
Meine Güte! Wie, meine Dame, möchtet Ihr sonst angesprochen werden?
Ok, joking aside. It's a convention, my lady. Using English you have that convention, too, addressing a noble woman like I did in the last sentence (I have no idea if this is still used in English nowadays). "Madame" is the French equivalent.
"Meine Dame" and "Mein Herr" is old fashioned in German, but we still use this construction when addressing a group of people in a formal way. That's not the case in English, but I guess different countries could develop different traditions ;)
In general, addressing someone with "mein" or "meine" adds the connotation of familiarity (or perhaps even friendliness). I would assume this is because e.g. saying "Mein Max" distinguishes Max from all other persons of that name in the sense that the person saying it knows Max quite well.
Instead of saying "Meine Damen und Herren" you can also say "Sehr geehrte Damen und Herren" which is more formal (and not at all to be taken literally).