The length of the two possible paths [one though each slit] to a given location on the screen typically differ. When they are the same, the probability of finding an electron at that location is relatively large, when they differ by 1/2 a cycle, they cancel perfectly (0 probability). When they differ by a whole cycle, the probability of will be large again. In most locations they differ by some other amount, so it varies from location to location smoothly. That is where the pattern comes from.
PS: When an outcome can happen more than one way, classically we add their probability together to get the probability of the outcome. However in QM we do something slightly different. We add their probability amplitudes. Unlike actual probabilities, probability amplitudes can be negative (or even complex). This is how you can have the sum of two paths give less probability than either path by itself (with the most shocking result when the probability amplitudes cancel giving 0 probability to find an electron).