Op, what is referred to as a particle in Modern Quantum Field Theory are excitations of the vacuum corresponding to simple poles in the 2-point correlators. These have a definite energy-momentum relationship (called a dispersion relation), remain stable for a definite time, and have various characteristics (mass, spin, El. charge, color, etc.). Even classical fields, such as the electromagnetic field, have single-particle excitations. For example, the photon is the one for the EM-field. Usually, the fields are named according to the name of the elementary particle that is their excitation.
These particles are not the billiard balls from Classical Mechanics. Feynman, as the co-founder of one of the more successful Qunatum field theories, Quantum Electrodynamics (QED), was certainly aware of this. Thus, he referres to particles in the above sense.
But, the video that you are referring to describes Quantum Mechanics of a single particle. In this case , one may construct a wave equation that describes the evolution of the particle. This wave is the wave of De Broglie's wave-particle duality hypothesis. It corresponds to the semi-classical (non-quantum) limit of the equation of motion for the quantum field, in an approximation where decays and collisions of the particle with others may be neglected, or the vacuum is different (the above coherent state).
