  We present the first results from the JWST ASPIRE program (A SPectroscopic
survey of biased halos In the Reionization Era). This program represents an
imaging and spectroscopic survey of 25 reionization-era quasars and their
environments by utilizing the unprecedented capabilities of NIRCam Wide Field
Slitless Spectroscopy (WFSS) mode. ASPIRE will deliver the largest
($\sim280~{\rm arcmin}^2$) galaxy redshift survey at 3-4 $\mu$m among JWST
Cycle-1 programs and provide extensive legacy values for studying the formation
of the earliest supermassive black holes (SMBHs), the assembly of galaxies,
early metal enrichment, and cosmic reionization. In this first ASPIRE paper, we
report the discovery of a filamentary structure traced by the luminous quasar
J0305-3150 and ten [OIII] emitters at $z=6.6$. This structure has a 3D galaxy
overdensity of $\delta_{\rm gal}=12.6$ over 637 cMpc$^3$, one of the most
overdense structures known in the early universe, and could eventually evolve
into a massive galaxy cluster. Together with existing VLT/MUSE and ALMA
observations of this field, our JWST observations reveal that J0305-3150 traces
a complex environment where both UV-bright and dusty galaxies are present, and
indicate that the early evolution of galaxies around the quasar is not
simultaneous. In addition, we discovered 31 [OIII] emitters in this field at
other redshifts, $5.3<z<6.7$, with half of them situated at $z\sim5.4$ and
$z\sim6.2$. This indicates that star-forming galaxies, such as [OIII] emitters,
are generally clustered at high redshifts. These discoveries demonstrate the
unparalleled redshift survey capabilities of NIRCam WFSS and the potential of
the full ASPIRE survey dataset.
