I've already seen some answers here: Why are diffracted wave fronts drawn with curved edges in textbook diagrams, when according to Huygen's principle they should be straight lines? and from what I've understood, with Huygens' original principle, you would have had just a straight wavefront propagating after the slit, as shown by the images in the post. Instead, with the Huygens-Fresnel principle, we have a curved wavefront near the edges of the obstacle. Why exactly do Fresnel's changes to Huygens' principle allow for this curved wavefront? Why doesn't a plane wave have that curve at its edges too?
Look at the introduction in Wikipedia:
The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction and also reflection. It states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms the wavefront.
The difference is that in the original Huygens principle
He assumed that the secondary waves traveled only in the "forward" direction and it is not explained in the theory why this is the case
The mathematical form of the secondary wavelets was used correctly by the addition of Fresnel, so the phases were kept and thus diffraction can happen.