All the lines and vertices have an amplitude. When you multiply the probability amplitude for the lines, the amplitude for the particles to go from wherever they start to wherever they meet, and to the next meeting point, and so on, and also multiply by the amplitude for each meeting point, you get a number that tells you the total amplitude for the particles to do what the diagram says they do. If you add up all these probability amplitudes over all the possible meeting points, and over all the starting and ending points with an appropriate weight, you get the total probability amplitude for a collision in a particle accelerator, which tells you the total probability of these particles to bounce off one another in any particular direction.