For the human players, the map has a data layer where every tile has a number. Tiles not on the track are 0 and tiles along the course are numbered in ascending order.
The game tracks the last numbered tile each player was on. When the players are too far apart so that they can't all fit on the same screen, then the players who have the smallest number get popped / killed. So if you cut a corner, then you're taking a risk -- you might be further ahead visually, but since you left the course, your last waypoint number will be lower than players still on the course. Those players can swerve in the other direction to force the issue and kill you, if they're paying attention.
In another data section, some of the numbered tiles are marked as key waypoints (my term, I don't know what the Codemaster devs called this). If you miss a key waypoint block entirely, then you are popped for cutting too much.
Those two mechanisms work together to encourage a kind of "iterated prisoners dilemma", on top of the normal racing mechanics. It's great fun.
There are some screenshots that illustrate this at https://bradders.org/MMs/ -- see the section titled "Cutting corners"
For AI players, there's a data layer telling them which way to go next on each tile, much like is described in the OP article here in MMs v1.