We present a detailed study of the partial rest-optical ($\lambda_{\mathrm{obs}} \approx 3600-5600\,\r{A}$) spectra of $N = 328$ star-forming galaxies at $0.6 < z < 1.0$ from the Large Early Galaxy Astrophysics Census (LEGA-C). We compare this sample with low-redshift ($z \sim 0$) galaxies from the Sloan Digital Sky Survey (SDSS), intermediate-redshift ($z \sim 1.6$) galaxies from the Fiber Multi-Object Spectrograph (FMOS)-COSMOS Survey, and high-redshift ($z \sim 2$) galaxies from the Keck

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... onic Structure Survey (KBSS). At a lookback time of $6-8\ \mathrm{Gyr}$, galaxies with stellar masses $\mathrm{log}(\mathrm{M_{\ast}/M_{\odot}}) > 10.25$ appear remarkably similar to $z \sim 0$ galaxies in terms of their nebular excitation, as measured using $\mathrm{[O\,III]}\lambda5008 / \mathrm{H}\beta$. There is some evidence that $0.6 < z < 1.0$ galaxies with lower $\mathrm{M_{\ast}}$ have higher $\mathrm{[O\,III]}\lambda5008 / \mathrm{H}\beta$ than $z \sim 0$ galaxies and are more similar to less evolved $z \sim 1.6$ and $z \sim 2$ galaxies, which are offset from the $z \sim 0$ locus at all $\mathrm{M_{\ast}}$. We explore the impact selection effects, contributions from active galactic nuclei, and variations in physical conditions (ionization parameter and gas-phase oxygen abundance) have on the apparent distribution of $\mathrm{[O\,III]}\lambda5008 / \mathrm{H}\beta$ and find somewhat higher ionization and lower enrichment in $0.6 < z < 1.0$ galaxies with lower $\mathrm{M_{\ast}}$ relative to $z \sim 0$ galaxies. We use new near-infrared spectroscopic observations of $N = 53$ LEGA-C galaxies to investigate other probes of enrichment and excitation. Our analysis demonstrates the importance of obtaining complete rest-optical spectra of galaxies in order to disentangle these effects.