This question is based, as pretty much everyone knows, on the book "QED, the strange theory of light and matter", by Feynman.
My answer is based on my limited understanding of the book.
Phase Arrows has a good description of Feynman's analogy.
As well the above source, if you visit Feynman Lecture Using Phase Arrows , you can watch RPF's description between times 29:41 and 36:27 of part He draws the "arrows" diagram on the chalkboard.
Similarly, if you visit Feynman Lecture On Photons, Feynman talks about this subject between times 59:33 and 60:32.
One of the differences between classical mechanics and quantum mechanics is that classically, objects have a definite path,or trajectory, so if you kick a soccer ball, you can be pretty certain where it will end up, and the path it will take.
Whereas in q.m.you can't say that a photon will definitely either follow a path or be found at a particular place. You can only assign a probability that a photon will be found at a particular place.
So Feynman sort of took this idea to the extreme, in a mathematical sense. Mathwise,you can say the photon took every available path, but all the paths that don't confirm to classical notions cancel each other out, by destructive interference. So you are left with just the paths that take the least time.
This is sometimes known as the sum over histories approach to quantum mechanics.