This one might actually be understandable! It is what Richard Feynman starts to explain at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
The difficulty is then proving that the total probability remains at 1, and maybe causality is hard too.
The path integral formulation can be seen as a generalization of the double-slit experiment to infinitely many slits.
Feynman first stared working it out for non-relativistic quantum mechanics, with the relativistic goal in mind, and only later on he attained the relativistic goal.
TODO why intuitively did he take that approach? Likely is makes it easier to add special relativity.
This approach more directly suggests the idea that quantum particles take all possible paths.
As mentioned at: physics.stackexchange.com/questions/212726/a-quantum-particle-moving-from-a-to-b-will-take-every-possible-path-from-a-to-b/212790#212790, classical gravity waves for example also "take all possible paths". This is just what waves look like they are doing.
Mentioned for example in quantum field theory in a nutshell by Anthony Zee (2010) page 8.
Articles by others on the same topic
The path integral formulation is a powerful framework used in quantum mechanics and quantum field theory, developed primarily by physicist Richard Feynman in the 1940s. It provides an alternative perspective to the conventional operator formulations of quantum mechanics, such as the Schrödinger and Heisenberg formulations. ### Basic Concepts 1.