What I am confused about is, what do fs and E physically represent, and what are they in an idealized battery model?
The force $\mathbf f_s$ is more commonly called the electromotive force (EMF). It is the external energy which is provided to the circuit. For example, in a battery it is the energy from the chemical reaction that pushes charges "against" the direction that it would naturally want to move (e.g. according to Ohm's law). In a battery the EMF is highly localized, it only exists at the surfaces of the electrodes. In the rest of the circuit the force on a charge is provided by the E field.
FYI, my personal opinion is that I do not like the equation $\mathbf f = \mathbf f_s + \mathbf E$ because it invites confusion like yours. The symbol $\mathbf f$ usually refers to a force with SI units of N and the symbol $\mathbf E$ usually refers to an E field with SI units of V/m = N/C, and the EMF here represented by $\mathbf f_s$ usually has SI units of V = J/C. So I cannot see how an author could expect students to not come away confused from reading this equation.
Regarding the "smooths out" comment. That is actually talking about the effect of surface charges. Consider a simple battery connected by two wires to a resistor. Without the wires, the terminals of the battery are slightly charged and produce an electric dipole field which varies over space. This field, like any dipole field, falls off rather rapidly away from the battery. However, when we attach the wires they obtain a surface charge which makes it so that the dipole field does not fall off so rapidly. In fact, it brings the full voltage of the battery smoothly over to the resistor. In the absence of the smoothing effect of the wires the potential across the resistor is miniscule, but with it the full voltage is smoothly applied across the resistor.