MOSFET channel length modulation - Glossing over physics
Remember, the drain contact is a reverse-biased P-N junction. There is a depletion region surrounding it that gets thicker with increasing reverse bias (increasing drain voltage).

This depletion region extends into the channel, and the electric field associated with it modifies the electric field being produced by the potential difference between the gate and the substrate, making it more difficult to achieve the inversion required to create the channel at that end. This is called "pinch-off" when it completely negates the inversion, creating a gap in the channel.

This paper: MOSFET Device Physics
and Operation contains a lot of the mathematical detail along with some nice diagrams.