A common misconception with the double slit experiment is that the particles are interfering with each other. In fact, the experiment is meant to show that considering photons as pointlike particles is only an approximation that in some cases breaks down spectacularly.
Rather than think about individual particles, think about a plane wave incident on the two slits (think about the wave being made of water if it helps). The amplitude (squared) of this wave corresponds to probabilities of finding photons - any photons, not just one or another. You wouldn't be surprised to see a water wave self-interfere when passing through slits, and indeed the wavefunction here does the same.
So where do the photons - the particles themselves - come in? Well, we don't detect infinitesimal amounts of energy everywhere, but rather a finite number of discrete packets. Their appearance is governed by the probability distribution of that wave. Going back to the water analogy, imagine you had an array of detectors that would trigger if the amplitude of the water wave became large, but that the triggering was probabilistic with greater probability coming from larger amplitudes. Depending on the detectors' sensitivity and the energy in the wave, you can expect a certain average rate of detections, but you can't say for certain where the next detection will be; you can only give a probability distribution. Nothing qualitatively changes if the wave amplitude is decreased to the point where you only get one detection on average every minute. The detections are temporally separated and don't interfere with one another - it is the underlying wave, which we are sampling with the detections - that is showing an interference pattern.