What is meant by the MOSFET is pinched-off? It is same as in the JFET case where the depletion region becomes too large until the electron cannot flow through the transistor?
When a positive gate-source voltage in NMOS is applied, As voltage increases once the voltage reaches the threshold voltage, the channel will be created and the device is ready to conduct electron or holes (in case of PMOS) through the channel between drain and source.
So by applying drain-source voltage it maximizes at the drain end and minimum at the source end. As Drain-source voltage is increased, it attracts the electrons from source end to drain end and current flows from Drain to Source, in this case, we're in the triode region.
As Vds increases, the channel becomes smaller till it's OFF, but in a real case, the channel is not completely pinched-off, in fact, it is more like pressing a pipe end with your fingers. There is a uniform electric field at the pinched end hence instead of stopping the current there is a constant saturated current.