An anode rod is not needed in some water heaters. But it that heater HAS an anode rod, it needs one. I can all but guarantee that they wouldn't have spent the money on placing one if it wasn't necessary.
True enough, many people do not know about them and do not understand how or why anode rods need to be maintained. They are a sacrificial lamb, so to speak. Their purpose is to prevent corrosion of the tank itself (or perhaps the electric heating element), and they do that by being eaten up instead of the tank being eaten up. When they eventually are used up, the tank then begins to degrade and will eventually start leaking as holes form.
For what it's worth, it looks like your electric heating element is damaged, and THAT is probably why you kept tripping the breaker.