Here is a perspective from optical engineering.
The major intuition comes from linearity.
Huygens principle says, the waves are superposition: the optical propagation is a linear system, and you can add up the waves to finish the propagation. Say the incoming wave is $x$, output wave $y$, the they are related as:
$$y = Ax$$
Where $A$ represents the propagation, and the specific form depends on propagation distance $z$ and wavelength $\lambda$.
One good thing about free space propagation is, it is space-invariant. Consequently you can express the system using convolutions: use a convolution kernel to represent the propagation. One example of this propagation kernel is the Rayleigh Sommerfeld formula. This simply says, matrix $A$ is a Toeplitz matrix, and can be diagonalized in Fourier domain. One eigenvector of $A$ is the planar wavefront, which has the form of the Fourier basis.
You can further simplify this relationship. For example if the propagation is large compared to wavelength, and paraxial, matrix $A$ reduces to the form of the Fresnel formula (a fractional Fourier matrix). In far field, it further reduces to the Fraunhofer formula, and $A$ is now a Fourier matrix.
So to summarize, Huygens principle formulates the propagation problem as a linear system. This greatly simplifies our model and helps us understand the optical waves, in an easy yet mostly accurate way.
Edits:
See this very helpful link http://www.mit.edu/~birge/diffraction/ for visualization of the above mentioned diffraction kernels.
