I will replace the previous answer to focus on the "subgiant" branch prior to red giant, rather than the pre-main-sequence or the "horizontal branch" of core helium fusion. Those are other times that the luminosity is constant, but this question is about the subgiant branch, which I missed before.
The reason the luminosity is nearly constant on the subgiant branch is related to the "mass-luminosity relation" of pre- and main-sequence stars. It is due to radiative diffusion and how it leads to a luminosity that depends only on mass, for a given composition. If you compare to pre-main-sequence tracks, you should find that the subgiants more or less retrace that prior evolution, just with a somewhat higher luminosity because many of the electrons have been swallowed into neutrons, reducing the opacity and increasing the all-important rate of radiative diffusion. It's essentially just a helium-dominated mass-luminosity relation, instead of hydrogen-dominated, as the radius rises due to the details of how the interior is evolving.
The reason the luminosity eventually rises steeply on the red giant branch is that as the degenerate core starts to build in mass, it starts to control the temperature of the fusing region, and this significantly changes the internal structure in ways that get into explaining red giants.