Feynman diagrams are labels for a way of breaking a complex integral into parts, each of which can be evaluated by applying fairly simple rules. As such they make it much easier to evaluate expressions which originally looked very difficult, and so make it possible for theoretical predictions to be made (and checked) much more quickly (and by people with lower technical skill levels). This led to a rush of progress in the theory and application of Quantum Electrodynamics, and paved the way for much more rapid progress in applying and testing other Quantum Field Theories (such as those used to model the strong and weak interactions of elementary particles).
Source: (1001) Alan Cooper’s answer to Why did Feynman diagrams revolutionize particle physics? – Quora