  We present emission-line measurements and physical interpretations for a
sample of 117 [OIII] emitting galaxies at $z=5.33-6.93$, using the first deep
JWST/NIRCam wide field slitless spectroscopic observations. Our 9.7-hour
integration is centered upon the $z=6.3$ quasar J0100+2802 -- the first of six
fields targeted by the EIGER survey -- and covers $\lambda=3-4$ microns. We
detect 133 [OIII] doublets, but merge pairs within $\approx$10 kpc and 600 km
s$^{-1}$, motivated by their small scale clustering excess. We detect H$\beta$
in 68 and H$\gamma$ emission in two galaxies. The galaxies are characterised by
a UV luminosity M$_{\rm UV}\sim-19.6$ ($-17.7$ to $-22.3$), stellar mass
~$10^8$ $(10^{6.8-10.1})$ M$_{\odot}$, H$\beta$ and [OIII] EWs $\approx$ 850
Angstrom (up to 3000 Angstrom), young ages (~100 Myr), a highly excited
interstellar medium ([OIII]/H$\beta\approx6$) and low dust attenuations. These
high EWs are very rare in the local Universe, but we show they are ubiquitous
at $z\sim6$ based on the measured number densities. The stacked spectrum
reveals H$\gamma$ and [OIII]$_{4364}$ which shows that the galaxies are
typically dust and metal poor (E(B-V)=0.1, 12+log(O/H)=7.4) with a high
electron temperature ($2\times10^4$ K) and a production efficiency of ionising
photons ($\xi_{\rm ion}=10^{25.3}$ Hz erg$^{-1}$). We further show the
existence of a strong mass-metallicity relation. The young highly ionising
stellar populations, moderately low metallicities, low dust attenuations and
high ionisation state in z~6 galaxies conspire to maximise the [OIII] output
from galaxies, yielding an [OIII] luminosity density at z~6 that is
significantly higher than at z~2, despite the order of magnitude decline in
cosmic star formation. Thus, [OIII] emission-line surveys with JWST prove a
highly efficient method to trace the galaxy density in the epoch of
reionization.
