this gif was at the top of r/4chan the other day (even though it seems like it's 8chan's work) http://i.imgur.com/BkkgY1g.gifv and it does a pretty good job of getting you up to speed.
Then you need live flight data. flightradar24 looks like a good bet, they even provide altitude information. The accuracy of that data can probably be massively improved with a Kalman filter.
It probably doesn't happen too often that there are three airplanes flying close to each other at a specific point in time. Even if there are, you might rule out all but one triplets of planes from the angles. You could also look at flights at different times (e.g. over the course of the last hour, how many flight paths have seen flights at 7, 20, 33 and 51 minutes apart?). That way they were able to identify which ray in the image corresponds to which plane and 3D flight path.
If you look at two intersecting lines, the angle will change depending on the view direction. They knew the angles of the 3D flight paths to each other, and the angles of the projections. To find the angle of the paths to the view direction you can use non-linear optimization. Or just do it manually in Blender (you only need two angles). You actually only need two rays.
I think the English word 'triangulate' is nowadays used for all kinds of geolocating based on cues from geometry, not just for the original method that actually involves triangles.