Plug one end of the HDMI cable into the HDMI Out port on your Xbox. Plug the other end into the HDMI In port on your capture card.
The Xbox console is designed as a . Its HDMI port is labeled "HDMI Out." Its job is to take raw video data generated by the GPU, encode it into a signal, and push it outward to a display (a TV or monitor). The Xbox is the speaker; it is shouting video data.
Affordable options like those from Anker or budget "USB Video Class" (UVC) sticks found on Amazon work well. HDMI Cable: A standard high-speed cable.