After spending some time researching about the centrifugal force, I now understand that it is needed in a non-inertial reference frame for Newton's Laws to hold true. However, I don't understand why we only feel the centrifugal force when moving in a circular path. For example, if you imagine yourself being spun around in a circle, then in your frame of reference you would feel yourself being pushed outwards. But since in your frame of reference you are stationary, the outwards force must be balanced by an inward pull (for you to remain stationary). So why would you only feel the outwards force but not the inwards force?
Suppose you have your feet pointing outwards and your head pointing towards the centre of rotation:
Now ask your self what direction the force you feel is pointing. Well, the force is pushing upwards on your feet, so the force points from your feet towards your head. In other words the force you feel is pointing inwards not outwards i.e. it is the centripetal force.
When you are standing upright imagine a downward external force on your head and an equal magnitude upward force on your feet.
These two forces will compress you and you “feel” being compressed as a result of these two forces acting on you.
Imagine that you are being spun round in an apparatus as shown in @JohnRennie ‘s diagram.
In his example there is an external force on you due to the “floor”.
In order for all parts of your body to have an inward force so as to rotate your body must be compressed.
So you “feel” compressed and interpret this as being acted upon by two forces one of which is real and causing the centripetal acceleration, the force of the the floor on you, and another force which is not real, an outward force which is supposedly helping to compress you - a centrifugal force.