It has already been said above that it is the picture of a particle that is at fault here. Observations should lead the model, and in this case, observations tell us that when we speak of a particle like a photon, we are not referring to a bb (i.e., some hard object with definite shape or even of zero size and definite trajectory). Hence the language leading to confusion: "the photon interferes with itself".
Let's use the photon and double slit thought experiment. In quantum mechanics, you can still think in terms of definite point particle paths (as Richard Feynman did -see Feynman Lectures in Physics Vol 3 ch.1-), but with the following caveats: (1) you must draw all the possible paths the particle can take from the starting point to the photographic plate, and
(2) to each path is associated a phase (this is the contribution of what is called the "wave nature of particles" in popular accounts).
You can consider just two types of path for simplicity: one path for the particle to go through one opening, the second path for the particle to go through the other opening. When the paths meet at some point on the photographic plate, they will have a relative phase. If the relative phase is zero (i.e., they are in phase), then this point on the plate is a possible point where the photon will end up. If they are exactly out of phase with each other at some other point on the plate, the photon will never end up at that point.
If you send a bunch of these photons one by one, they will appear most often at the points on the plate where the two paths were perfectly in phase, never appear at points on the plate where the paths were exactly out of phase, and will appear somewhat often at points where the two paths were "somewhat in phase" (between exactly out of phase and perfectly in phase).
Those are the
Rules of Quantum Mechanics in terms of a point particle picture. If you ask *why* particles behave according to these rules, *that* we don't know. But if you accept these rules, we can explain and predict the behavior of more complicated systems in terms of them.
Now, I mentioned that you can view quantum mechanics as 'point particles taking all paths possible', and this was a common view even by the developers of what is now quantum field theory. However, this is not the modern picture of what a particle is. Modern quantum field theory does not have point particles as the fundamental entities zipping around. Rather, something known as a "quantum field" is viewed as the more fundamental entity (don't attach any metaphysical meaning to 'entity' here). Particles are excitations ("quanta") of these quantum fields (and as a consequence are what we described as quantum particles above). When you calculate how e.g. an electron will interact with another electron electromagnetically, you start with two electrons (never mind that they are quanta of their own field, an "electron field")...then having a "photon field" present, you can compute how the electrons will will interact with each other via the photon field. If you want you can say that the electrons interact via fundamental units of the photon field: photons are exchanged in all possible ways and numbers between the electrons.
The notion of a qauntum field, its quanta, and their interactions is subtle conceptually, and the best way to understand it, unfortunately, is to study it. But perhaps all that's important is the following: new rules were needed in the early 1900's for how things like electrons act fundamentally (and therefore how we view what a 'particle' is). Then we later found out that to describe interactions among these particles required newer rules that again modified our view of what particles were. The modern view is given in terms of quantum field theory.
As a final note, these fields should not be confused with classical fields like the electromagnetic field. An electromagnetic field that exists in some region of space is a quantum state of a bunch of photons (I say quantum state to emphasize the fact that the collection is not a bunch of individual bbs...in fact the number of photons is not a definite thing for this macroscopic field).
