It is not a literal translation of the Italian sentence, which should have been:
- È la mia lettera. = It's my letter.
This sentence replies to the question "What is this?".
- La lettera è la mia. = (literally) The letter is the one that belongs to me/has been written by me (→ rephrased) It's my letter.
This replies to "Whose letter is this one?", inferring that there are several letters, and one of them belongs to or has been written by the speaker.
- La lettera è mia. = The letter is mine.
This replies to "Whose letter is it?", inferring that there is only one letter being spoken of.