If you're just curious about why the instantaneous system of forces has a horizontal component which leads to horizontal motion (w.r.t. the earth), then you can consider that the system is *instantaneously* similar to a reverse pendulum. If the friction doesn't allow any slip between floor and rocking chair, then the centre of mass is *instantaneously* constrained to move on a tangential motion around the pivot on the floor, as in the following diagram:

[![Force system][1]][1]

The pivot is the black dot; the centre of mass is the blue dot; its constrained instantaneous direction of motion is the dashed red curve.

This situation implies that the reaction force of the floor is directed as the green vector $\pmb{F}_{\text{R}}$ in the picture, and its magnitude must balance the component of the gravitational force $\pmb{F}_{g} \equiv m\pmb{g}$ along the line connecting centre of mass and pivot. With some trigonometry we find that this magnitude is $\lvert \pmb{F}_{\text{R}} \rvert = \lvert \pmb{F}_g\rvert\ \cos\theta$, in terms of the angle $\theta$ depicted in the picture.

So the total force on the centre of mass is $\pmb{F}_{\text{T}} = \pmb{F}_{g} + \pmb{F}_{\text{R}}$, and again with some trigonometry we see that by construction it must be tangential to the dashed red curve. It therefore has a vertical component and a horizontal component:

[![Total force][2]][2]



  [1]: https://i.sstatic.net/dQam8.png
  [2]: https://i.sstatic.net/63sUO.png