See the picture. At the start of the throwing motion the spur of the atlatl is inside the cup in the rear of the dart. As the atlatlist pulls the dart and atlatl forward the point end of the dart starts to rise, and the spur tip begins its rotation out of the cup.
As the atlatl handle is pulled forward of the atlatlist’s face, the rear end/spur end of the atlatl is, at the same time, levered upward. The spur is soon completely out of the dart’s cup and the butt of the dart is riding on the top face of the spur. The dart is connected to the spur only because the atlatl is moving faster forward than the dart which is flexing as it moves forward, in a vain attempt to get out of the way of the spur.
At the moment of the throwing motion when the atlatlist’s arm is fully extended forward and he or she begins the rapid downward wrist flick, the dart butt will slide along and off the top face of the spur provided there is nothing to get in its way.
In an earlier article about spurs, I advised and continue to advise that the tip of the spur should be a blunt rounded shape that would easily rotate into and out of the cup in the rear of the dart. Also, I said that there should be no extra length of wood behind the spur that might come down on top of and hit the rear end of the dart during the wrist flick motion, and that the spur itself should not have any bulky wrappings around it that the dart but might get hung up on.
I also advised and continue to advise that the top face of the spur should be flat or have a groove in it to avoid the dart slipping off one side or the other of the spur during the throwing motion.
As you can see from the drawing, during the throwing motion, the tip of the spur rotates into and out of the the rear of the dart, and the rear of the dart ends up sliding over the top face of the spur until the atlatl has pushed the dart away completely. It is there for necessary for the spur design to allow the dart as frictionless a disengagement from the spur as possible.