Say in a given instance a car moves with speed $v$ and consider any wheel of the car. How fast is it going? Is it the case that the center of the wheel moves at the same speed as the car i.e. $v$? Why is that?
The wheel is connected to the car through an axle which goes through the center of the circle that is our wheel. The point which is in contact with the ground is the instantaneous centre of rotation and that very point on the wheel has a speed of zero. Clearly the center of the wheel has non-zero velocity.
So in any instant the movement is a rotation about a fixed axis, where the fixed axis in the considered instant lies in the contact point with the ground and the speed is zero on the fixed axis. If the angular velocity of the wheel is $\omega$ and the wheel has radius $R$, then the speed at the center of the wheel is $v_C = \omega R$.
Why is this speed $v_C = \omega R$ the same as the speed $v$ of the car? Maybe it is self-evident but I really do not see why.