Diffraction effects depend on the wavelength of the light. Considering a single narrow slit with monochromatic light, light with wavelengths much larger than the slit will not be transmitted and light with wavelengths much shorter than the slit will be transmitted without significant diffraction effects, but light with wavelengths comparable to the slit will show significant diffraction effects.
The reason that diffraction effects are able to split white light into its different colors is because white light is composed of an incoherent combination of many different wavelengths of light. The different wavelengths get diffracted by different amounts, and the effect you see is that the white light gets split into its spectrum of colors. Additionally, since the light is incoherent, you don't see dark and bright spots like you would with monochromatic light.
How do we understand from Huygen's principle that light with wavelengths much shorter than the slit do not diffract very much? This is because points near the middle of the slit and points near the edges of the slit, which are both emitting spherical waves will interfere destructively except for in the direction straight ahead.