Original response: There is a perfectly valid physical explanation: use Maxwell's equations to find how the wave propagates beyond the aperture. Use as boundary conditions the idea that the wave is completely killed off beyond the aperture. Sound reasonable? This results in an integral. Turns out that the integral can be interpreted, at least to a very good approximation, as Huygens's principle!
So it's a clean principle - derivable from Maxwell's equations which are the foundation of electrodynamics, and is not some contrived idea.
October 2016 addition: Looking at my answer after so long, I see that the original question asked for an "intuitive" picture of the origin of wavelets. The fact is that even the basic Kirchoff integral is scalar in nature while the fields are vectorial; the reference given in my comments below (Jackson) does attempt to extend this to the full vector case. However, this is non-trivial to do and also to understand.
Nevertheless, the basic idea is that the simplest "Green function" needed to solve the equation has a form implying "spherical wavelets emanating from every source point". Unfortunately, the simplest Green function is not universally applicable to all problems, but needs to be different for and must be adapted to every particular boundary condition.
Reading Jackson carefully it is evident that there are several different approximations that can be made under various circumstances, and the popular idea of Huygens's principle is the best simple summary of a mathematically complicated situation. And now I admit I've exhausted my own capacity to clarify this problem and its solution.