The ``pinch-off region'' (or ``saturation region'') refers to operation of a FET with more than a few volts. Look at Figs. 5.2 and 5.10 in Eggleston. In this regime, the current is almost independent of (for a given ). What's happening inside the transistor is that the depletion regions due to the biases have enlarged to allow only a narrow path for the current (hence the ``pinch-off'' nomenclature). In this regime, as increases, it increases the length of the depletion zone, which increases the resistance, basically compensating for increased current due to increased voltage-- so the current remains nearly constant. The particular constant value depends on , which sets the width of the depletion zone. This is the normal operating regime of a FET, in which current going through depends on the control gate-to-source voltage .
(You can also have a complete pinch-off when the bias voltage is so large that the entire channel is non-conductive and you get basically no current.)