There is a classic physics experiment called the double-slit experiment. A photon source (i.e. light) is shot at a barrier where there are two slit openings. A screen behind the barrier shows the light that makes it through the slits. The pattern of the light on the screen has various spots of higher and lower intensity. Thus demonstrating that light has the same properties of a wave, namely the ability to constructively and destructively interfere.

Some time later, we came up with other experiments that demonstrated that light has properties that behaved more like a particle. This led to a problem. The classical understanding of physics didn't allow for light to behave as both a wave and a particle..

One possible solution to the problem, proposed by physicist Richard Feynman, suggests that instead of treating the light traveling a single path from the emitter to the screen, it travels an infinite number of paths, every possible combination from emitter to screen. The final result is the sum effect of all those infinite paths (hence the integral part of the name). This is not a physical explanation of the phenomenon, but a mathematical one to explain why there would be an interference pattern from something that acts as both a wave and a particle.