For a MOSFET:
Source at ground, drain voltage is Vd, and gate voltage is Vg.
With Vd=0, as Vg increases passed the threshold voltage VT, an inversion layer charge is developed under the gate and connects the drain and source with a resistive region that conducts proportional to the inversion charge. It acts like a resistor in the first order.
As Vd increases, current increases because the channel is conducting like a resistor. It is nononlinear, of course, because as Vd increases, the voltage along the channel is increasing and there is less voltage from gate to channel and thus the charge in the channel is decreasing (from source to drain). That is why, the IV curve is not a straight line, but bows down as Vd increases.
At some point, Vd is high enough that the voltage from gate to drain is less than VT, thus the inversion layer is not supported at the drain. This is the "pinch off" point. The voltage at that point (the drain end of the channel) can no longer increase. The transistor is operating in what is called the "saturation" region. Vdoltage at the pinch off point no longer changes, so the current Vpinch/R (where R is inversely proportional to the charge in the channel) is fixed.
In reality, there is a lot more going on beyond this simple explanation. For example, physical location of the pinch off point is modulated by Vd so that is Vd increases, the inversion region (the channel) gets shorter and thus resistance gets lower and the current goes up. This is called "channel length modulation."