Theyre recording a demo in the engine.
Bring down the console and do
record your_demo_name
you can name the demo whatever you wish. This is saved into the id1 folder if youre playing vanilla Quake. Otherwise itll be in the mod folder of whatever mod youre playing.
Then, to get the nice smooth look as seen in
this video, run the demo file through
this tool.
Just drag and drop the
.dem file onto the tool.
Then to replay it in the engine, put the file into the id1 folder (or the folder for whatever mod you recorded it in), open your engine and do
playdemo your_demo_name
from there youll need to use OBS or some other screen capture software to create the video.