Written in the typical, beautiful Feynman style, this book is fine for an advanced student who already knows quantum mechanics and Green functions from a standard source like Sakurai or Merzbacher. It presents Feynman's interpretation of quantum mechanics in chapter 1 via the two-slit experiment, and the rest of the book is devoted to showing how to formulate and calculate the one particle Green function for simple systems, systems with completely integrable classical analogs (it's implicitly assumed that Ldt is a closed differential, where L is the classical Lagrangian). The path integral formulation was also later used by other researchers to arrive at a semi-classical approximation to the three body problem, a nonintegrable and even chaotic classical system (nonintegrable classical systems cannot be solved by the standard method of finding a complete set of commuting constants of the motion).
The functional integral formulation of Brownian motion was formulated earlier by Norbert Wiener. An analogous formulation of quantum theory was arrived at independently by Feynman, who took seriously a conjecture by Dirac about the meaning of the exponential of the classical action as a probability amplitude. A more complete treatment of classical Brownian motion (including the so-called 'Feynman-Kac formula' for Brownian motion) was given later by Mark Kac in "Probability and Related Methods in the Physical Sciences".
Chapter one presents with Feynman's interpretation of quantum mechanics, the interpretation accepted by theorists today, as nonclassical rules for combining probability amplitudes for particle propagation. Waves are not mentioned because the mental gyrations inherent in the Copenhagen 'wave-particle duality' are completely avoided in the Dirac-Feynman approach. See, as forerunner of Feynman's interpretation, Dirac's discussion of photons interfering with themselves in a hypothetical two-slit experiment, in the introduction to his famous text "Quantum Mechanics".
In other words, this book is for students who are ready to face the fact that there is no 'wave-particle' picture, or any geometrical picture of reality, at the quantum level: the reader who really understands Feynman's description of the two-slit experiment will realize that we cannot say about the hydrogen atom that an electron is moving about the nucleus, unless we do a scattering experiment to detect the electron (an electron doesn't follow a path, nor is it in two different places at the same time, there is in the end only the space-time propagation of quantized fields). As Feynman admitted, we do not really 'understand' quantum mechanics, although we can do all of the calculations describing experiments. The 'measurement problem', the Einstein-Podolsky-Rosen paper and subsequent experiments and papers on quantum teleportation make this viewpoint clear. Quantum mechanics, nature at the microscopic level, is stranger than anything that you can imagine!
The Dirac-Feynman interpretation of quantum theory is presented by Sakurai, who also discusses the measurement problem. Merzbacher doesn't teach Dirac-Feynman but does discuss Galilean invariance via gauge transformations, and sets up the two-body problem in a form that is useful for understanding the enstein-Podolsky-Rosen paper.