Why does the MOSFET drain current increase abruptly when Gate voltage is greater than the threshold voltage?
Instead why does the current increase abruptly from zero when a threshold voltage is hit?


It doesn't. It is (very near) zero below threshold, but it only starts increasing gradually from zero as it goes above threshold. Here's a "typical" characteristic for a low-voltage MOSFET:




  Why is the sudden increase in carrier concentration?


It's not the carrier concentration that changes at threshold. It's the availability of the channel.

Below threshold, the channel is "pinched off" so that no current can flow, even though there are charge carriers available.

Imagine you use your fingers to pinch a drinking straw closed. If you slowly release the pressure of your fingers, you see no current flow until you get to some "threshold" pressure and water starts flowing. It's not because the amount of available water changed, it's because you allowed a path to open so that the water that was there all along could flow. The channel of the MOSFET behaves similar to that.