Translation:I have another daughter, but she is older.
I thought the same thing. Just always think of it in terms of "What sounds better in English?" Although sometimes when I do this I get it wrong. It really is about exposing yourself to as much Italian as possible and letting it sink in. Remember, no one learns anything by not making mistakes. It is part of the process. Take guesses and see if you are right. If you are not then try again. CIAO!
I have just seen a similar discussion in the Dutch course because Dutch and German use "bigger" with children in much the same way as Italian does. Which happens to be the opposite of how English uses "little" with children. A "little" child is small, young and immature all at the same time, with none of these aspects really dominating. Similarly, for a child to be "più grande" means to be bigger, older and more mature all at the same time. The best one-word approximation in English is to say "older" because it sort of implies the other two.
Just like in English, the apostrophe indicates that a word is shortened by one or more letters. In this case, una altra was contracted to un'altra, removing the a of una. I don't know if this particular contraction is obligatory, but it is certainly very common because una altra is a bit inconvenient to pronounce with its clash of two a's.