What should I do with my back arm while I fence, and why do fencers hold it up like that?
Keep it relaxed and out of the way, usually raised behind you with the upper arm roughly level and the hand loose, and never use it to block, grab or cover anything. The raised pose is a leftover from dueling days, when the free hand helped defend and fencers kept it high and away from the opponent's point. Modern rules ban it from defending at all, but the habit stayed because it still helps your balance and keeps the arm clear of the target.
Think of the back arm as a counterweight. In a lunge, swinging it down and straight back along the line of your rear leg helps you drive forward and keeps your shoulders steady. On the recovery, bringing it back up helps pull you back into guard. Not everyone uses the classic pose: many sabre fencers rest the back hand on the hip or hold it low behind them, and plenty of épée fencers let it hang loosely. That is all fine. What matters is that it stays relaxed, behind you and in the same place every time. A tense, hunched shoulder just wastes energy over a long day.
The rules are the same in all three weapons on the big point. Using your non-weapon hand or arm to attack or defend, such as pushing the blade aside, catching it or blocking with your forearm, is a red card, and any touch you scored in that action is annulled. Grabbing your body cord or any other electrical equipment with your free hand during the bout is a yellow card. You also cannot switch the weapon to your other hand mid-bout unless the referee allows it because of an injury. Putting your free hand down on the strip while you duck is allowed.
Covering is where the weapons differ. In foil the back arm is off target, so if it drifts across your chest or you twist so it shields valid target, that is a yellow card for covering, your touch is annulled, and the referee can still award a hit that landed on the arm in place of the target. In sabre and épée the back arm is valid target, so a hit there simply counts. Whatever your weapon, practice footwork and slow bouts with a coach or partner calling out every time the arm creeps forward, because a reflexive hand block under pressure is an easy way to give away a red card.