How do you explain diffraction using Huygens principle?

The Huygens-Fresnel principle posits that every point on a wave front is a source of new spherical wave fronts. In other words, you can imagine that at every point along the crests of a traveling wave (the parallel blue lines in the figure below), new wave fronts are generated at each point, which propagate out spherically in every direction, like ripples on a pond.
The (linear superposition, or sum) of those spherical wave fronts leads to the formation of new wave fronts at later times. If a beam of light is not clipped by any aperture, long straight wave fronts will lead to more long straight wave fronts. This is because the spherical wave fronts only interfere constructively when they have all traveled the same distance as each other, namely one wavelength away from the original wave front at a time that is one wave period later.
However, if the beam is clipped, some of the spherical wave fronts are interrupted. This leads to the interference pattern shown by the grey semicircles in the figure above. At the points where those grey contours cross each other, and along the lines joining those points, there will be constructive interference. This is nicely illustrated by the animated gif below
For an excellent video discussion of this effect, see this episode of Veritasium:
The most relevant part of the video is at 4:14, although the whole thing is worth watching.