Hooke's law says the force is linear with the displacement, but that only works when the force and the displacement remain in alignment. You can have a long thin cantilever and if the tip force is axial, the global force vs displacement curve will be linear.
Rotate that force 90 degrees so the tip force is causing bending in the beam. As the tip rotation increases, the elements rotate, and the direction of the force changes from a bending force and goes toward an axial force, especially for the elements near the tip. Also, as the tip foreshortens and moves closer to the base, the bending moment on the elements at the base is reduced. Both of these changes increase the apparent stiffness of the global structure, but no material stiffness had to change, it is purely a result of large deflection and large rotation. That is why it is called geometric nonlinearity.