The key thing you need to understand is that the centrifugal force depends on the reference frame. In some reference frames it exists, in others it doesn't.
Let's take a simple example of a planet orbiting a star at a fixed distance. Before we can analyze their motion, we need to choose a reference frame.
We can choose an inertial reference frame, where there is no centrifugal force. In this frame, the planet moves in a circle. There is one force on the planet - the gravity of the star, which acts as a centripetal force keeping the planet moving in a circle.
Or we can choose a rotating frame, which is centered on the star and rotates with a period equal to the orbital period. In this frame, the planet does not move at all. Its position is completely fixed. There are two forces acting on it - gravity towards the star, and centrifugal force away from the axis of rotation of the frame (which, by construction, means away from the star). The two forces cancel out, so the net acceleration of the planet is 0 (consistent with the observation that it doesn't move at all).
When analyzing more complex systems, we must remember that the centrifugal force applies differently to different objects (the acceleration on each is directed away from the axis of the frame, with magnitude proportional to the distance from the axis); and that there is an additional force, the Coriolis force, which acts on objects which are moving in this frame. (It doesn't come into play in the previous example, because nothing moves.)
In general, whenever you have a system where rotation is involved, you have a choice how to analyze it - either from an inertial frame, or from a rotating frame which matches the system's behavior. Each will lead to correct results, possibly in a different way. But you must be careful not to confuse the two, and apply in one frame logic that belongs to the other.