Does grass grow faster after cutting?

Well, that's a simple question with a complicated answer.

Plants require a number of "inputs" in order to grow. Most importantly, Carbon dioxide (CO2), light and water. They also require an appropriate temperature (this temperature ranges and varies depending on the plant) in order to metabolize these inputs to turn them into a usable source of energy. (The human body does the same thing... but uses much more complex mechanisms)

I'm assuming you are living in a warm climate where the grass does in fact grow year round (but slows down in the winter)... or perhaps it goes brown and "dormant" in the winter. Both are the physiological response to an environmental trigger within the plant. This is most likely a result of cooler temperatures, shorter exposure to sun (shorter days), and a poorer light angle (the earth tilts, and the light hits the plant at a different angle).