How do light interference patterns correspond to the particle nature of light? Interference patterns like those produced by a two-slit experiment make sense to me when I imagine light as a wave, with peaks cancelling out troughs in some locations and two peaks adding together in other locations. What doesn't make sense to me is how the particle nature of light is explained in this instance. How does a phase shift result in a lack of photons striking some spots and more photons striking others?
 A: The particle nature of light comes up when you are detecting the photon. It always appears localised at some point. This can be seen in double slit experiments done by sending one photon at a time. This has been done but I wasn’t able to find images (thank you @annav). So here is the image of the experiment done which shows the accumulation of detections over time. 

So now that we know photons come in discrete chunks (as they are localised on our detectors) why are there regions on the detector where no photons go? Well that is because the probability (amplitude) distribution of the photon has wavelike properties. This is what quantum mechanics tells us. This means that each photon interferes with itself. 
A: They don't. Interference patterns are a product of waves, not (classical) particles. The particle nature of light is not explained by interference patterns.
Young's slits and the resulting interference pattern are not evidence of the corpuscular theory, as once the particle theory was called. Interference patterns were taken as direct evidence that light is a wave not a particle. 
That we now understand something about photons exhibiting wave-particle duality does not mean that any particular/specific result of an experiment that supports one interpretation of wave-particle duality has to support the other.

How does a phase shift result in a lack of photons striking some spots and more photons striking others?

It tells the obsever that photons exhibit wave characterstics, that they are in some way waves that interfere with themselves.
