Why frictional force helps an object to move in a circular path?

When the car is driving in a straight line the tire need the friction of the road to gain traction and move the car (the tires will spin on ice). On a curve the tires need the friction of the road both for traction and to avoid sliding off the road. When the brakes are applied either on a straight section or a curve the friction of the road results in a force that stops the car. As you suggest, if you try to push the car with its brakes applied the friction stops the car from moving. The main difference is that the car can move because the tires roll and the only energy lost is due to rolling friction (deformation of the tires) which is modest.