If centrifugal force is a pseudo-force then why we can observe its effects outside of rotating frame?

When objects fly off a spinning table, from an external reference frame they simply stop travelling in circles. At the point of "unsticking" they do not accelerate radially outwards from the centre of rotation but merely stop accelerating inwards; they continue in a straight line tangential to the circle and at right angles to the radial direction. Thus, no centrifugal force is present - indeed, it is the very absence of any force subsequent to breakaway which explains their behaviour.

When you look at the rotating table or the centrifuge from outside you are not seeing the centrifugal force, but only Newton's first law, that inertial objects continue to move in a straight line.

Suppose that using an inertial frame of reference you observe a body undergoing a centripetal acceleration because of a force which is acting on it.
Using the inertial frame you can apply Newton's laws of motion to predict what is happening to the body and that force acting on the body has a Newton third law twin (action/reaction).

Now choose a frame of reference which is not moving relative to the body which is undergoing a centripetal acceleration when observed in the inertial frame.
What do you observe?
The body is not accelerating (moving) relative to the non-inertial frame and yet it has a force acting on it, the force that caused the centripetal acceleration in the inertial frame.
This immediately means that Newton's second law does not work $F\ne m\,0$.
So that Newton's laws can be used in the non-inertial frame a fictitious/pseudo force is added so that the net force on the body is zero and now Newton's second law does predict the correct acceleration of the bod, zero, in the non-inertial frame $F-F_{\rm pseudo} = m\,0$.
This fictitious force does not have a Newton third law twin but enables Newton's second law to be used in the non-inertial frame.

In a sense you have looked at the situation the wrong way round.
The pseudo force is a way of making Newton's law work in a non-inertial frame but there is a real force which is there in both the inertial and non-inertial frame which causes the centrpetal acceleration of the body when observed in the inertial frame.