I've read about feedback loops, how much this steady state error is for a given gain and what to do to remove this steady state error (add integral and/or derivative gains to the controller), but I don't understand at all why this steady state error occurs in the first place.

If I understand how a proportional control works correctly, the output is equal to the current output plus the error, multiplied by the proportional gain (Kp). However, wouldn't the error slowly diminish over time as it is added (reaching 0 at infinite time), not have a steady state error? From my confusion, it seems I'm completely misunderstanding how it works - a proper explanation of how this steady state error eventuates would be fantastic.