I learned the rules of chess from a children's book in third grade, but have never played a full game as far as I can remember. Yet this situation seems extremely obvious to me. One of the most fundamental rules from my children's book was that you cannot make a move that puts yourself in check. Moving k to the right would clearly do that. And it's pretty obvious when you think through what would happen if k moved to the right. R would capture k, which would not put R in check, because the game would be over.