Most videos like this are recorded with capture hardware that will record the outgoing video and audio feed from the xbox or other console and record what is going out to the television in real time.
A common brand of capture hardware is
Hauppauge, which were kind enough to provide a
diagram of how the setup would normally look.
Just alerting you now, quality recording equipment can be pricy, especially if you aren't aware of what you are getting into. My personal recommendation is to straight up ask someone who is doing a video like this what their equipment recommendation would be. It can be a bit daunting at first, and I don't want to give the wrong impression by not letting you know.