We investigate what drives the redshift evolution of the typical electron density ($n_e$) in star-forming galaxies, using a sample of 140 galaxies drawn primarily from KMOS$^{rm 3D}$ ($0.6lesssim{z}lesssim{2.6}$) and 471 galaxies from SAMI ($z<0.113$). We select galaxies that do not show evidence of AGN activity or outflows, to constrain the average conditions within H II regions. Measurements of the [SII]$lambda$6716/[SII]$lambda$6731 ratio in four redshift bins indicate that the local $n_e$ in the line-emitting material decreases from 187$^{+140}_{-132}$ cm$^{-3}$ at $zsim$ 2.2 to 32$^{+4}_{-9}$ cm$^{-3}$ at $zsim$ 0; consistent with previous results. We use the H$alpha$ luminosity to estimate the root-mean-square (rms) $n_e$ averaged over the volumes of star-forming disks at each redshift. The local and volume-averaged $n_e$ evolve at similar rates, hinting that the volume filling factor of the line-emitting gas may be approximately constant across $0lesssim{z}lesssim{2.6}$. The KMOS$^{rm 3D}$ and SAMI galaxies follow a roughly monotonic trend between $n_e$ and star formation rate, but the KMOS$^{rm 3D}$ galaxies have systematically higher $n_e$ than the SAMI galaxies at fixed offset from the star-forming main sequence, suggesting a link between the $n_e$ evolution and the evolving main sequence normalization. We quantitatively test potential drivers of the density evolution and find that $n_e$(rms) $simeq{n_{H_2}}$, suggesting that the elevated $n_e$ in high-$z$ H II regions could plausibly be the direct result of higher densities in the parent molecular clouds. There is also tentative evidence that $n_e$ could be influenced by the balance between stellar feedback, which drives the expansion of H II regions, and the ambient pressure, which resists their expansion.
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