The "centrifugal" force experienced by an object moving along a curved path is only present in the rotating reference frame of the object. The fact that it does not exist unless you are in the rotating frame is evident from an analysis of Newton's laws.
Newton's First Law (N1L) - An object at rest tends to stay at rest and an object undergoing uniform linear motion will continue in that motion along a straight line unless acted upon by an outside force provided one is in an inertial reference frame.
This is important for the understanding of rotation because it says that something moves in a straight line at constant velocity (both how fast and the direction of motion) unless some other thing exerts a force on it. That is, you need two different bodies to change the state of motion. An inertial reference frame is best defined as a reference frame where Newton's First Law is valid. That is to say, things do not change velocity unless there is an obvious external agent. Strictly speaking, Earth is not an inertial reference frame because things fall to the floor unless some force holds them up. It is obvious to us that gravity pulls the object down, but this is only true because we live in a post-Newton time.
Newton's Second Law (N2L) - The total acceleration an object experiences is equal to the force exerted on that object divided by the mass of the object. Further the acceleration and the force point in the same direction.
What does this mean for the current discussion? First, in order for an object to accelerate there must be a non-zero force on it. Also, the direction of the acceleration is the direction of the force. So, by looking at the acceleration of an object we can see which direction the net force points.
Newton's Third Law (N3L)- If one object (A) exerts a force on a second object (B) then B exerts a force equal in magnitude and opposite in direction on A.
The third law is the least understood law. At first glance it would seem to suggest that there is no net force because forces show up in equal and opposite pairs. Well, if there are two forces equal in strength but pointing in opposite directions then the net force should be zero, and there should be no acceleration (N2L), right? Not quite. The third law works on two different bodies while the second law is concerned with the forces on a single body.
So, what does this have to do with anything? First, Newton's laws are only valid in an inertial reference frame or in a frame where you can explicitly consider all or safely neglect some of the external forces. Due to gravity, Earth is not an inertial frame, but most of the time we can neglect the gravitational force, especially if we confine ourselves to a horizontal plane. In a rotation problem/experiment the only inertial frame is outside of the rotation.
Think of it this way, one person is on a spinning merry-go-round (Mary) and another is standing on the ground (Stan). Stan is closer to an inertial frame than Mary because he can see N1L working on Mary. On the other hand, Mary is not in an inertial frame because she 'feels a force pushing' her out from the center with no other object exerting it. That is, for Mary N1L is not valid. Stan's view is the one we need to take when analyzing the problem.
Stan sees that Mary's velocity is constantly changing. She may be moving with the same speed the whole way around, but her direction of motion is always changing. If you think about it a little you should be able to convince yourself that Mary's velocity direction is changing toward the center of the path. This is the direction of her acceleration. By N2L we see that the acceleration is in the same direction as the force, and Stan sees that there is a net force on Mary. Further, this force is directed toward the center of the circle, not away.
As for N3L, the centrifugal force cannot be exerted on Mary because the centripetal force leads to an acceleration toward the center. If the centrifugal force where the third law partner of centripetal, Mary's acceleration would be zero. This is not the case.
From Mary's point of view everything is jumbled. She experiences a force pushing things away from the center of rotation. If she where on a huge merry-go-round (10 km radius) totally enclosed with no way to look outside, she would be hard pressed to determine from a local experiment whether she were rotating or just on a tipped platform.
Centripetal force is a true force in the sense that a separate object must exert it. That object can be a rope (like in tether-ball), a mass (like the Earth holding the Moon in orbit), or a charge (like the positive nucleus holding the electrons to the atom). Centrifugal force is a 'fictitious' in the sense that it is caused by an object's own inertia. The former exists in an inertial reference frame and is easily understood from Newton's Laws, while the latter is an artifact of a non-inertial reference frame and requires considerable effort to understand where it comes from.