We present the first characterization of the Gunn–Peterson trough in high-redshift galaxies using public James Webb Space Telescope (JWST) NIRSpec spectroscopy. This enables us to derive the first galaxy-based intergalactic medium (IGM) opacity measurements at the end of reionization. Using galaxy spectra has several advantages over quasar spectra: it enables measurements of the IGM opacity in any extragalactic field over a continuous redshift range $4\lesssim z\lesssim 7$, as well as measurements of the intrinsic Lyman $\beta$ opacity. Our novel constraints are in good agreement with state-of-the-art ground-based quasar Lyman $\alpha$ forest observations, and will become competitive as the number of JWST $z>5$ galaxy spectra rapidly increases. We also provide the first constraints on the uncontaminated Lyman $\beta$ opacity at $5 \lt z \lt 6$. Finally, we demonstrate the power of JWST to connect the ionization state of the IGM to the sources of reionization in a single extragalactic field. We show that a previously reported galaxy overdensity and an excess of Lyman $\alpha$ emitters detected with JWST in GOODS-South at $z=5.8\!-\!5.9$ coincides with an anomalously low IGM opacity to Lyman $\alpha$ at this redshift. The local photoionization rate excess can be fully accounted for by the cumulative ionizing output of $M_{\rm {UV}}\lesssim -10$ galaxies in the overdensity, provided they have $\log _{10}\langle \xi _{\rm {ion}} f_{\rm {esc}} / \ [\rm {erg}^{-1}\rm {Hz}]\rangle \simeq 25$ (e.g. $\log _{10}\xi _{\rm {ion}} / \ [\rm {erg}^{-1}\rm {Hz}]=25.4$ and $f_{\rm {esc}}=40~{{\ \rm per\ cent}}$). Overall, this breakthrough offers a new way to connect the galaxy large-scale structure to the state of the IGM, potentially enabling us to precisely identify the sources of reionization.