For all this to make sense it's important that Newton's three laws of motion are assumed to be true, in the sense that a force and space (and a straight line) and time is defined the same way in both frames.
To those in the inertial frame, the (pseudo) centrifugal force is obviously equal and opposite to the centripetal force. Since if the centripetal force $F$ allows for 0 acceleration in some rest frame, then removing it would create a force $-F$.
Circular rotation frame
Imagine you were in the rotating frame (like a carousel) with everything enclosed within it rotating at the same rate. There is no way of proving that another person resting on the frame of the carousel is not at rest in a straight line. Remember, they can't see what is outside the carousel. Assuming this, you would 'observe' other peculiar phenomenon that are not needed in the inertial frame.
To show this, you could draw an arbitrary point on the frame of the carousel with a marker and call that your reference point. Since the frame is taken to be a rest in a straight line, the point is not accelerating. Yet any object in free fall on the carousel accelerates relative to that point. It just so happens that the acceleration points directly away from the centre of the carousel. Any object that is at rest must therefore be acted upon by some other reaction force towards the centre (another arbitrary point), with equal magnitude. (This force is what is observed as centripetal in the inertial frame)
What is the magnitude of that force? Well, to the people stuck inside the carousel, there's no way to tell that it depends on the angular velocity because they do not know that they are spinning. They can only find that the force is proportional to mass because acceleration doesn't change, and proportional to $r$ by experiment and deduction.
Why can't the people in the rotating frame just assume they are rotating? Well, they can and it would make their physics simpler (fewer unexplained laws). The problem is, they have no way of knowing which is right unless they observe what happens once they break the perpetual circular motion.