He ought to be using what Wikipedia calls a "perspective projection", or, sufficiently in this case, a restriction thereof. Notice how the edges of walls do not appear straight on the screen, even though he mentioned correcting "a fisheye effect" (tellingly alluding to a popular tutorial on this topic at lodev.org).
A correct formula is on line 485 of my implementation linked above, found the old fashioned way using basic geometry.
On my old laptop, the demo looked fine. But after switching over to a widescreen (and higher FPS) you can tell that it's slightly wrong up close, and noticeably wrong (jittery) in the distance.
A correct formula is on line 485 of my implementation linked above, found the old fashioned way using basic geometry.