Original: Feynman diagrams were originally discovered by Feynman, by trial and error, as a way to represent the contribution to the S-matrix from different classes of particle trajectories.
Simplified: The points in Feynman diagrams where the lines meet represent two or more particles that happen to be at the same point in space at the same time. The lines in a Feynman diagram represent the probability amplitude for a particle to go from one place to another.