Please refer any text book for details. Here in brief:
You apply a potential to the gate. The gate accepts some charge depending on its capacitance value. Some initial current must flow to charge this capacitor.
Imagine a traditional capacitor with two plates. The two plates have equal charges (but opposite in sign) and it creates an electric field within the dielectric.
In the case of the MOSFET, the substrate is the other electrode of the capacitor. A very high electric field will be setup because of the small gap between the electrodes. (E=v/d)
A significant amount of the electric field is taken over by the insulator but the effect will be felt in the depletion layer (it can also be an enhancement mode). That will modulate the current between the drain and source.
Basically the electric field modulates the number of carriers in this zone. Theoretically speaking, the electron band structure gets distorted by the presence of the strong electric field. As you know, only the top of the band structure is relevant and a small distorting can cause a large effect.