Why is the car moving if the unbalanced force is zero?

This is a great example of where physics and our gut feelings about physics don't match up. This is a very common conceptional difficulty students have and is often jumped on by examiners. The best way to think about it is with an example. Let's imagine you have a car with a mass that is being pushed forward by a force of 500N and is being pushed in the opposite direction by a force of 500N. The unbalanced force is, of course, zero. Now if we remember Newton's second law which says that F=ma we can figure out what this means for the acceleration. We know that F=0N and that m is not zero. Therefore to keep the equation balanced a must be zero. What does it mean to say a=0? It actually means two things. Either the car is not moving or it is moving but not accelerating or decelerating (we call this "constant velocity"). Therefore Newtons second law tells us that a car can be moving without an unbalanced force acting on it. 

A simpler, but less rigorous, way to think about it is to remember what a force actually is. A force, in very simple terms, is something that changes the motion of an object. If there is no net force acting on an object its motion won't change. So if it is moving with a constant speed and no net force acts on it the object will continue with it's motion.