I assume that the VAR system had some competent coders involved, who know basic computer vision techniques, when they were developing the system. However, they clearly didn't use it the other day.

One particularly common technique is to do what is called "image rectification". Essentially, it takes lines that are know to be parallel in real life (in this case the painted lines on the edge of the field and the 18 yard box) and performs a mathematical operation to "undistort" the image so that the perspective projection is eliminated and the lines are parallel in the new image. This simplifies many tasks in computer vision like finding corresponding points in two images, merge images from multiple vantage points, etc. It can also be used to look at relative positions like in the VAR offside problem. You can see a pretty good tutorial of how this is done at https://boracchi.faculty.polimi.it/teaching/IAS/Rectification/demo_affine_rectification.html

Because we have a 3D vantage point and a single camera, we can make the rectification do two things:
1. cause parallel lines to really become parallel
2. choose either one set of parallel lines to be horizontal lines or the other set to be vertical
NOTE: If we had two cameras with the same vantage point, we could have done another step in the rectification process and "undistorted" to ensure that the two sets of parallel lines were also orthogonal, but that can't be done easily with a single camera.

Here are the results. The conclusion is that the Uruguay player was very, very far offsides. Furthermore, it isn't obvious to me that the VAR system is performing image rectification. It is possible that they are using an algorithm like this, finding the offside position, then reversing the affine transformation to find the line in the unmanipulated frame. Even if that were the case, someone screwed up when they were picking the offside line.

https://imgur.com/a/8MElfrg