Feynman diagram by Ciro Santilli 34 Updated Created
I think they are a tool to calculate the probability of different types of particle decays and particle collision outcomes. TODO Minimal example of that.
And they can be derived from a more complete quantum electrodynamics formulation via perturbation theory.
At Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), an intuitive explanation of them in termes of sum of products of propagators is given.