There are three reasons that people right-handed people wear it on the left arm and that left-handed people wear it on their right arm.
(1) This is the main reason now. Sit at a table or desk. Get out a piece of paper to write with. To keep the paper stationary, right-handed people almost always use their left hand to hold the paper down (vise versa with left-handed people). Thus if you wear the watch on the arm which is holding down your work, you can always see the time without moving any body part. It is great for homework or business paperwork. It is especially useful if you are in a hurry. (This is the same reason most people wear their watch face pointing out. Otherwise the watch face would be scratching the tabletop while writing and cannot be seen).
(2) You are more likely to be using your favored hand for other uses - and it is too busy to lift to see the time of day. Your opposite arm is less used, and thus is free to lift and see the watch.
(3) Winding watches. This is becoming a moot point, but winding a watch is much easier when it is worn on the opposite arm.