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We present emission-line measurements and physical interpretations for a sample of 117 [O iii ] emitting galaxies at z = 5.33–6.93, using the first deep JWST/NIRCam wide-field slitless spectroscopic observations. Our 9.7 hr integration is centered upon the z = 6.3 quasar J0100+2802—the first of six fields targeted by the EIGER survey—and covers λ = 3–4 μ m. We detect 133 [O iii ] doublets, but close pairs motivated by their small scale clustering excess. The galaxies are characterized by a UV luminosity M _UV ∼ −19.6 (−17.7 to −22.3), stellar mass ∼10 ^8 (10 ^6.8−10.1 ) M _⊙ , H β and [O iii ] _4960+5008 EWs ≈ 850 Å (up to 3000 Å), young ages, a highly excited interstellar medium, and low dust attenuations. These high EWs are very rare in the local universe, but we show they are ubiquitous at z ∼ 6 based on the measured number densities. The stacked spectrum reveals H γ and [O iii ] _4364 , which shows that the galaxies are typically dust- and metal-poor (E ( B − V ) = 0.1, $12+\mathrm{log}({\rm{O}}/{\rm{H}})=7.4$ ) with a high electron temperature (2 × 10 ^4 K) and a production efficiency of ionizing photons ( ξ _ion = 10 ^25.3 Hz erg ^−1 ). We further show the existence of a strong mass–metallicity relation. The properties of the stars and gas in z ∼ 6 galaxies conspire to maximize the [O iii ] output from galaxies, yielding an [O iii ] luminosity density at z ≈ 6 that is significantly higher than that at z ≈ 2. Thus, [O iii ] emission-line surveys with JWST prove a highly efficient method to trace the galaxy density in the Epoch of Reionization.