If the fictitious forces (ex: centrifugal) help the observer feel stationary, why does the person feel a force in the opposite direction?

What force an object is experiencing depends upon the reference frame of the observer. If you sit in the rock's frame (the one which is rotating), the only force acting on it (excluding frictional forces at contact) is fictitious centrifugal force which directs outwards away from the disk's centre.

Generally speaking, fictitious force comes into the picture when you go from an inertial frame to a non-inertial one.

So in stationary frame, how do you explain the rock moving away since outward force is not there? Observing in stationary frame, you see the rock follows percievably a different path. What keeps the rock rotating is a force (centripetal) directed towards the centre. This can be provided by the friction between rock and disk surface. If this force suddenly vanishes, the rock will follow a straight line motion. This motion when viewed in rotating frame appear as if the rock is moving away from the centre.

Draw a tangent to the disk at a point. the rock has tendency to move along such a tangent at each instant(due to its inertia). hence it feels as though it is pushed outwards w.r.t. the disk. and for the same reason it needs a force along the radial direction to keep up with the disk.