How is the gate capacitance of a MOSFET taken care of by a high current?
Capacitance is measured in farads, which are equivalent to coulombs per volt. This means that a given capacitance can contain a certain number of coulombs of charge per volt of electromotive force applied to the capacitance.

Current is measured in amperes, which are equivalent to coulombs per second. This means that a given current moves a certain number of coulombs of charge per second.

When we apply a larger current to a capacitance, we move a larger number of coulombs per second, thereby increasing or depleting the charge on the capacitance faster.