I'm currently deriving the gravitational redshift in the Schwarzschild spacetime by following Wald's dicussion on the topic.
I came across this video, which talks about the gravitational redshift through, what the author of the video calls ''covector wave fields''. I've never heard of that and have never seen it in any book, what does it mean?
The video is simply talking about the wave covector, i.e. the more natural form of what is commonly called the wave-vector; see my answer here defining this. A covector is just another name for a 1-form. It is properly a covector field since there is a covector at each position (hence the dependence on $x^\mu$ in my other answer). Any gravitational physics book should cover the basics of this (e.g. "Gravitation" by Misner, Thorne and Wheeler).