There are many types of physical devices that can act as inertia dampers: Ballast. Brakes - kinetic energy redirected as heating due to friction between surfaces pressed together.

Could inertial dampeners exist?

No such process is known to exist in the real world: if current understanding of physics is correct, such a process would be impossible. There is currently no known material or technology that is able to eliminate or negate the effects of inertia that all objects with mass possess.

How could inertial dampeners work?

Inertial damping system, inertial dampers, or inertial dampeners, were a system used on almost all starships to counter the effects of rapid acceleration and deceleration by sustaining and absorbing the natural inertia of a vessel as it moved through space or if it was under attack from another vessel.