How does the particle in the double slit experiment interact with the slits?

Most particles do hit that space, and are blocked. The source of particles must be such that it emits in all directions. Often, something is done to simulate a point source. An ideal point source emits in all directions. For example, in the case of light, a single slit is put in place before the double slit. The single slit simulates a point source and radiates in all forward directions.

Update after @Jaywalker's answer

@Jaywalker brings up an important aspect that I glossed over: How to reconcile the wave picture and the particle picture? How can a single "particle" (say, an electron) be launched toward both slits? The answer depends on what we think about when we say "particle". If we mean a tiny little object that moves through space like a projectile, then the answer is "it can't". This is primary evidence that that picture of "particle" is not a good description of what is going on. In short, all "particles", including electrons and photons, are quantized excitations of a field. Energy and momentum are transferred at particular locations, giving the impression that a object hit something. The field exists everywhere in space except where space is occupied by an obstruction. The excitation of the field (the "particle") exists everywhere, but the interactions occur at particular locations. The field obeys some wave equation, and thus exhibits interference.