Okay, so, the first thing to understand is that the textbook¹ is trying to get you to understand a theory put forward by Christiaan Huygens in the late 1600s. It turns out he was (generally) right about the wave nature of light, but the actual specifics are iffy. Don't get too bent out of shape trying to make everything make sense, because... well, it doesn't. It was, however, a good enough approximation to work out basic rules of optics, and to be improved by later scientists.
Then, the second thing. You are, it turns out, very confused. :) There is no "curl" — in fact, the secondary wavelets can be thought of as full spheres. And we don't get to plane waves from this at all (I'll explain that in a second.)
Waves propagate out from their source in the same shape as the
source. If it starts from a big, flat, plane in 3D space, you get a plane
wave. When the source is a point in 3D space, you get a sphere. Huygen's idea was that, if you took such a sphere of light at a certain time, you could figure out where it would expand to next by pretending that each point around that sphere² is actually a new point source, and then considering the edge of the wavefront in the future to be the boundary of all of those spheres combined.
This is useful, because when the light is moving through the same substance ("isotropic homogenous"), it moves at the same speed, keeping its shape. But light moves at different speeds in different materials. Newton's model — light as particles — doesn't give a satisfying answer for why light bends when this happens. But this does! There's a great animation here, and here's a less-clear still visualization from Wikipedia:
As the light hits the — let's say water — the "secondary wavelets" travel more slowly in the new medium. Because of the angle of incidence, the wavefront doesn't strike the water all at once. And if you draw a line across the front of all the little circles that do occur at the same time (in this picture, the first green line is drawn at the time the right edge strikes the water), you can see that the wavefront is now aligned differently.
So, this model demonstrates why light bends when it goes from air to glass, which is of course fundamental to photography as we've known it for the past two hundred years. But don't get too hung up on understanding the specifics of the actual wavefronts and secondary wavelets (unless you really need it for the test), because ***those are just the model, not really a correct representation of what happens in the real world.
This model also explains diffraction — how light gets bent around corners:
I think that's basically self-explanatory — the new (imaginary) point sources at the aperture give us rays going off in unexpected directions (making our images fuzzy).
And — plane waves. This isn't really specific to Huygens' theory, but a) a plane wave is easier to think about than a spherical one, which is why it's used in examples, and b) if you're far enough from a point source, from a practical point of view, you can treat the wave as plane wave — just as we know the Earth is a sphere, but because it's so big, we can treat the ground as flat for most purposes.
So, finally, how is this relevant to photography?
Having a basic model for how refraction works is pretty useful. And while this 400-year-old version is not perfect, it's actually reasonably consistent with complete, modern electronmagnatic theory for practical use. (For more on that, I definitely refer you to Physics Stack Exchange.
Likewise, it explains diffraction. (And, reflection, for that matter.)
The thing about far away spheres approximating straight lines is useful, because we generally consider sunlight to have parallel rays. (See for example What's the technical difference between artificial and natural light?).
And finally, but not mentioned, mostly because I don't follow the math (you can try here, or, again, I'll refer you to Physics Stack Exchange), Huygens' principle has an interesting repercussion — unlike when you throw a stone in a pond, the waveforms don't diffuse and get thicker as they propagate. This means that if someone turns a distant light on for exactly a second, you observe it for one second as well, just later. And if that weren't true, we couldn't get clear pictures of stars!
So, finally — do you need to know this to take photographs? Absolutely not. You don't even need to know more surface-level explanations of why refraction works — just the very basics of "hey, a lens focuses light". But, I think that knowing these things can help understanding in a wide variety of photographic situations. That's why there's a "theory of light" section in your textbook, after all (even if it is a unfortunately confusing).
1. Sample pages here, by the way. This is on page 27 of the textbook (page 46 of the PDF, as there is a lot of front-matter.
2. And there are an infinite number of points, which the diagram does not make obvious