Every way an event can happen is represented by a little arrow whose length and direction are set by rules; the final probability is the square of the length of the arrow formed by adding all the contributing arrows head to tail. The arrow for a path rotates like a fast stopwatch hand as the light travels, so paths of different lengths point in different directions.
To find how likely light is to go from one point to another, you must add the arrows for every conceivable path it could take, not just the straight one. Nearby paths with nearly the same length have arrows pointing the same way and reinforce, while paths far from the shortest one cancel out, which is why light appears to travel in straight lines only as an approximation.
All the phenomena of light and electrons reduce to three elementary actions: a photon travels from place to place, an electron travels from place to place, and an electron emits or absorbs a photon. Each action has an arrow computed by a simple rule, and every complex event is assembled by combining these three.
The strength of the electron-photon interaction is captured by a single measured number near 0.085424, whose origin no theory explains; physicists just put it in by hand to match experiment. Feynman frames this unexplained constant as a lingering embarrassment at the heart of an otherwise precise theory.
Treating light as if it secretly tries every path, including going slower or faster than light and bouncing off every point of a mirror, reproduces the real behavior once the arrows are summed. The familiar laws of reflection and lenses emerge not as rules light obeys but as the net result of countless canceling alternatives.
When light hits glass and some photons reflect while others pass through, there is no detectable property distinguishing which photon does what; the theory predicts only the probability, not the individual outcome. Feynman insists this indeterminacy is not ignorance to be cured later but an accurate description of how nature behaves.