You can explain diffraction (and refraction!) from classical wave mechanics; you don't need quantum mechanics really. Let's try a hand-waving explanation.
A very important theorem to understand how waves propagate is the Huygens-Fresnel principle. It says that the wave propagation can be seen as a perpetual absorption at all points everywhere, and these points re-emit the absorbed wave as spherical wave. The wikipedia article shows nice pictures of those re-emitting points.
When the wave passes through a hole (see this picture), the center of the wave stays flat thanks to the contribution of central emitting points. However, the sides of the wave are bent, because there are no more points to "complete" the wave. That's basically where diffraction comes from.
Now, let's see how the wavelength matters. For that, we need to understand the phase of waves. It represents the relative temporal delay between waves of the same wavelength. For example, two waves which oscillate at the same time are said in phase. Waves in phase add up, whereas waves in anti-phase cancel each other. Back to diffraction: we observe what is going on in front of the hole. If you put a detector in front of the hole, sufficiently far away, it will be more or less at the same distance from all the emitting points of the hole. Thus, all the spherical waves will be in phase, so they add up, i.e. there is a wave. Now, let's put the detector on the side. The emitting points will not be all at the same distance to the detector: the waves they emit will all be mismatched with each other (they have a phase offset). As a consequence, these wave cancel each other so the total wave is dimmer on the side.
When you do the same reasoning with a longer wavelength, the result changes. You place the detector on the side as we did before. The different emitting points will still be at different distances from the detector. But now, they oscillate much slower, meaning that they don't get so much time mismatch (=phase offset) and don't cancel out. As a consequence, the side-going wave is not so dim. This explains why long wavelengths diffract more.