  We present a first sample of 117 [OIII]$\lambda\lambda$4960,5008-selected
star-forming galaxies at $5.33 < z < 6.93$ detected in JWST/NIRCam 3.5$\mu$m
slitless spectroscopy of a $6.5 \times 3.4$ arcmin$^2$ field centered on the
hyperluminous quasar SDSS J0100+2802, obtained as part of the EIGER
(Emission-line galaxies and Intergalactic Gas in the Epoch of Reionization)
survey. Three prominent galaxy overdensities are observed, one of them at the
redshift of the quasar. Galaxies are found within 200 pkpc and 105 km s$^{-1}$
of four known metal absorption-line systems in this redshift range. We focus on
the role of the galaxies in ionizing the surrounding intergalactic medium (IGM)
during the later stages of cosmic reionization and construct the mean
Ly$\alpha$ and Ly$\beta$ transmission as a function of distance from the
galaxies. At the lowest redshifts in our study, $5.3 < z < 5.7$, the IGM
transmission rises monotonically with distance from the galaxies. This is as
expected when galaxies reside at peaks in the overdensity field of an IGM that
is ionized by more or less uniform ionizing background, and has been seen at
lower redshifts. In contrast, at $5.7 < z < 6.14$, the transmission of both
Ly$\alpha$ and Ly$\beta$ first increases with distance, but then peaks at a
distance of 5 cMpc before declining. This peak in transmission is qualitatively
similar to that seen (albeit at smaller distances and higher redshifts) in the
THESAN simulations. Finally, in the region $6.15 < z < 6.26$ where the
additional ionizing radiation from the quasar dominates, the monotonic increase
in transmission with distance is re-established. This result is interpreted to
represent evidence that the transmission of the IGM at $z \sim 5.9$ towards
J0100+2802 results from the ``local'' ionizing radiation of galaxies that
dominates over the much reduced cosmic background.
