Sure, that quote is from Feynman’s lecture on the Principle of Least Action.
In quantum mechanics, Feynman showed in his Ph.D. thesis and later wrote the book quoted with Feynman and Hibbs (which is the easiest one…
Sure, that quote is from Feynman’s lecture on the Principle of Least Action. It was only about classical physics. The justification for that statement comes from the application of the variational principle onto the action. You can think of this as adding a small deviation parameter to the action functional and then minimizing deviations, thus minimizing the path.
In quantum mechanics, Feynman showed in his Ph.D. thesis and later wrote the book quoted with Feynman and Hibbs (which is the easiest one to read about this) that the quantum path integral only obeys this principle probablistically. Instead, each infinitesimal section of path obeys the infinitesimal unitary Schroedinger operator which comes from solving Schroedinger’s equation. This provides probabilities for each path with maximal probability at the minimal action path.
This probability function is like a normal distribution (if you go to imaginary time) for a free particle with Planck’s constant acting as the square standard deviation. As you take Planck’s constant to zero, the normal distribution becomes more and more peaked until it converges to a Dirac delta function distribution on the minimal action path. The Dirac delta function is a distribution with infinite height, width zero, and integral 1, so it is a probability function that selects out a single variable. Taking Planck’s constant to zero represents the transition from quantum to classical physics, hence why this derivation is consistent with the principle of least action in classical physics. It was a hugely beautiful result that led to the development of quantum field theory and Feynman diagrams.