Overview—Where we're going
Before we start, here is a brief "road map" of where we're going. The first sections are all explanations of "classical" Physics: that is, the understanding of the 19th century world, before the introduction of quantum mechanics. These provide a critical background for understanding the 20th century changes. Don't skip or skim them "to jump to the good stuff" because without this background, the good stuff won't make any sense!
What is Light? and When Light Waves Meet explain that light is a "wave," and exactly what that means, and introduce the critical concept of interference which explains what happens when two light waves meet. The Slit Experiments In Classical Physics introduces the "Young Double-Slit Experiment," an empirical validation of all the theories that were discussed in the first two sections. The Slit Experiments Repeated with M&Ms shows how hard objects behave in a similar experiment. In doing so, it sets up the crucial dichotomy between "waves" (such as light) and "particles" (such as M&Ms) and the important ways in which they act differently. You're Deliberately Wasting My Time makes these ideas explicit. So by the time you reach that point in the paper, you know a lot about 19th century Physics. You know what a wave is, and what a particle is, and how they act differently (particularly with respect to "interference"), and how the double-slit experiment validates all these theories. And (this is critical!) you should find the concepts intuitive: not easy, by any means, but within your grasp. Then we jump into the 20th century, and quantum mechanics.
New Information Comes to Light gives the first suggestion of some holes in the 19th century view of the world. It introduces the key notion of a "photon," a particle of light. The Double Slit Experiment Revisited shows how you can take the exact same experiment that validated the wave theory of light before, add one tiny new wrinkle (photons), and the whole thing gets turned on its ear. This is the critical turning point in the paper. If you've followed everything up to this point, then you should read this section with shock and amazement; because it shows actual experimental results that fly completely in the face of the "intuitive" picture of the universe that you've just built up. Groping Toward a Theory, Putting it All Together, and One More Experiment show you how, from those startling results, you can begin...