Building on a comment by CuriousOne (who, honestly, should leave off commenting since he only ever writes answers in then anyway):
A photon is not an object in and of itself. A photon is an excitation in a quantum field, which is not localized but fills space. In the double slit experiment you have an emitting source, a mask with two slits, and a viewing/recording screen, and all of those are immersed in quantum fields. In fact, all the individual particles that make up the emitter, mask and screen and all the force mediation between them are also excitations of various fields, but let's not be too distracted here: keep your eyes on the photon.
The field can be roughly visualized like waves on the surface of water, but the crests and troughs measure probability rather than height. If you have a tank of water with a double slit mask and put some kind of oscillator on one side, it causes waves that travel through the slits and cause a diffraction pattern on the other side. Search "water wave diffraction" in your favorite engine and you'll find a lot of images of water-wave double slit experiments, and even many of ocean waves diffracting around various landmasses.
Now imagine you have a very large experiment, or a very small research partner, and your research partner is surfing on the waves in your experiment. Because he can only surf near a crest, the crests of the waves directly correlate to the places where you will probably find your research partner at any given time that you measure his position. Of course, your wave emitter emits a spherical wave and your partner doesn't have to be right on a crest, so you could find him anywhere in the tank... But there are some places where you are more likely to find him.
So you set out with your experiment: your research partner surfs over the waves and marks where he lands each time. Of course, since you only have two small openings in the sea wall, most of the marks are going to be on the sea wall; but when he does come ashore, the marks he makes there over time will show a distinct interference pattern, because he is following waves that show a distinct interference pattern.
What's important to note here though is that the interference pattern is not generated by the waves; it is revealed by the waves, and is generated by the geometry of the experimental setup. A single slit would show a different pattern, as would a diffraction grating, etc. even though the source emits a circular wave each time. The change in geometry causes a change in the final measurement.
Same with photons. The photons are excitations of the field, and the field contains a probability distribution that behaves as a wave. Some places are very likely to contain an excitation, and some are not very likely, and of course there is a continuous spectrum between those extremes. And the way those probabilities distribute is governed by the wave-like nature of the field.
If you mix up the concepts, you might say "the photon interferes with itself and causes an interference pattern." This is incorrect because a single photon does not set up a pattern. If you send many photons, however, they do show a pattern: and that pattern reveals the probability shape of the experimental setup, which is solely a function of the geometry of the experiment. More accurately it could be said that "the photons' distribution follows the probability distribution of their field."