Why the gear engage when engine is running?

The gear engages when the engine is running because the engine provides the power needed to move the vehicle. The transmission's job is to transfer that engine power to the wheels. If you tried to engage a gear with the engine off, you'd be attempting to move the car solely by the force you apply to the gear shifter – which is not enough to overcome the resistance of the transmission and the vehicle's weight. The engine's rotational force, transmitted through the engaged gear, is what makes it possible to turn the wheels.