The screen does trigger the collapse of the particle wave function to a specific point on the screen. The probability of collapse at a specific point is given by the magnitude squared of the wave function at the screen. Thus, when one carries experiment many times (shooting one particle after another), an interference pattern appears, in the form of darker areas where more particles have collapsed, and lighter areas where the probability of finding a particle is low.
However, as long as the place, where the collapse occurs on the screen, does not allow us to distinguish through which slit the particle has passed, the interference fringes are preserved.