Best way to understand the need for saturation region can be understood using voltage-transfer characteristics (VTC).
View attachment 94424View attachment 94425
Fig. 1 consists of basic structure of CS amplifier while Fig2 shows its VTC.
We want LINEAR amplification of a signal i.e. the amplified signal should be linearly related to input signal so that we get distortion free output. U can see in Fig2 that this linear region(corresponding to amplifier characteristics) is from pt. A to pt. B where NMOS is in saturation region. If u bias NMOS in triode or linear region, as u want;-), u would get a highly distorted output.
Ahead, as above, similar to other members explanation, MOS can be understood to be acting like a current source(with some parallel resistance to it) in saturation region causing its i-v curve slope to be almost constant. While in triode region it acts as a voltage-controlled resistor where the slope is not constant & changes with the applied input voltage. Actually, from this behaviour the VTC arises which is pretty clear.
Hope u will get clear now.