All matter has both wave and particle properties, for small particles it is possible to observe the wave properties. For massive particles it is impossible to observe the wave properties, the theoretical wavelength is too short. The reason particles have wave properties is because all matter interacts with each other thru the EM field, i.e when a particle is colliding with another the electrons are the first to "see" or "sense" each other (there is something called virtual photons). Because the interaction is in the EM it has to have a wavelength/wavefunction associated with it. The term interference is historical, the slit experiment for photons showed a pattern similar to what water waves do. Even single photons "interfere" but it is better to think of an allowed path explanation (Feynman) where the brights spots occur. The electron will choose one slit at random, with this new path it will look to interact with the screen at a point that works with its wave function, this causes the INTENSITY pattern (note: I did not want to use the word "interference"!):