The technique can detect the intersection point of a (fixed size, axially aligned) moving bounding box and a hull defined as a BSP tree. In this sense it is rather limited compared to newer methods but it works rather well and the implementation is pretty robust, and fast enough for the primitive hardware of the day.
Hello, video creator here. This is indeed the actual code, and it is written in QuakeC, a language created specifically for the game logic in Quake. The part I'm not sure about is why the creators wanted to add in extra side beams at all.