The trick is, centrifugal force is a fictitious force.
Centrifugal force exists! To everyone denying it, do this to them: xkcd.com/123. But it is a fictitious force. To quote wikipedia:
A fictitious force is an apparent force that acts on all masses whose motion is described using a non-inertial frame of reference, such as a rotating reference frame.
So, if you sit in a merry-go-round, you can feel a force pulling you out. You can measure it. For you, this force exists. But for your mother standing outside the merry-go-round, watching you, there is no centrifugal force. She can see the merry-go-round applying a centripetal force to you, so you go along with the merry-go-round and not fall off. If it didn't your mass makes you go in a straight line and you fall off.
The reason why the two observers observe different forces is, that the merry-go-round is not a inertial frame of reference. Where the ground, your mother stands on, is.
In a inertial frame of reference there is no centrifugal force. But in a non-inertial frame of reference there might be.
So the centrifugal force appears to be there because an observer in a merry-go-round is not in an inertial frame of reference. By changing frames of reference you can eliminate it.
