Most of the hydrogen in the intergalactic medium (IGM) was rapidly ionized at high-redshifts. While observations have established that reionization occurred, observational constraints on the emissivity of ionizing photons at high-redshift remains elusive. Here, we present a new analysis of the Low-redshift Lyman Continuum Survey (LzLCS) and archival observations, a combined sample of 89 star-forming galaxies at redshifts near 0.3 with Hubble Space Telescope observations of their ionizing continua (or Lyman Continuum, LyC). We find a strong (6σ significant) inverse correlation between the continuum slope at 1550 Å (defined as F λ ∝ λ β 1550 obs ) and both the LyC escape fraction ( f esc,LyC ) and f esc,LyC times the ionizing photon production efficiency (ξ ion ). On average, galaxies with redder continuum slopes have smaller f esc,LyC than galaxies with bluer slopes due to the higher dust attenuation in redder galaxies. More than 5% (20%) of the LyC emission escapes galaxies with β 1550 obs < −2.1 (-2.6). We find strong correlations between β 1550 obs and the gas-phase ionization ([O iii]/[O ii] flux ratio; at 7.5σ significance), galaxy stellar mass (at 5.9σ), the gas-phase metallicity (at 4.6σ), and the observed FUV absolute magnitude (at 3.4σ). Using previous observations of β 1550 obs at high-redshift, we estimate the evolution of f esc,LyC with both redshift and galaxy magnitude. The LzLCS observations suggest that fainter and lower mass galaxies dominate the ionizing photon budget at higher redshift, possibly due to their rapidly evolving metal and dust content. Finally, we use our correlation between β 1550 obs and f esc,LyC × ξ ion to predict the ionizing emissivity of galaxies during the epoch of reionization. Our estimated emissivities match IGM observations, and suggest that star-forming galaxies emit sufficient LyC photons into the IGM to exceed recombinations near redshifts of 7-8.