I would most likely say "the letter he sent me" and leave out "which".

Both of these are fine. a) 'the letter which he sent to me' - a little more formal b) 'the letter which he sent me' - a little less formal c) Stephen's suggestion: 'the letter he sent me' - even less formal. All of these are acceptable.

I think B is more common in US English, A moreso in UK English. Grammatically both are correct.

All three options ok. Why? I think to attempt to answer "why?" would mean that someone would analyse the sentence for you and show you the structure. In itself, it would not provide a "why", except that these are all used and considered correct.