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Furthermore, why are the two first answers wrong?

Maybe it's here?

https://moodlearchive.epfl.ch/2020-2021/pluginfile.php/2739416/mod_resource/content/5/Delineation.pdf

starting from page 49.

But i'm still very confused about this part.

This question should be understood in the context of delineation. We have shapes (think edges of objects) in the image and we wish to extract them in a useful manner (as some mathematical/programmatic objects we can operate on and reason about). One such object can be a straight line, which has a simple equation but is not very expressive - many edges are curved and we wouldn't be able to describe them with lines. For this reason we may want to use splines, which are a bigger family of shapes (they can be curved, but you can also make a spline become a line, with properly chosen parameters). However, they are still limited in that they are topologically equivalent to lines - without going into topology (I'm not an expert myself), this means that they come in "one piece" - they have a beginning and end, you can make the two coincide if you want to have a closed shape, but you can't represent edges of two separate objects (such as the cup and sunglasses from slide 49) with one spline. You'd have to have two splines, but then it becomes a discrete, two-step problem of 1) choosing the right number of splines (and initializing them) 2) optimizing the spline shapes.

The level set approach is a way to define a yet broader class of objects which can represent multiple "pieces" on the image and the number of those pieces is controlled by adjusting continuous parameters (so that we can find the number of those pieces by continuous optimization). The idea is that we define a function z = f(x, y, t) [as opposed to splines which are defined as (x, y) = f(t), 0 <= t <= 1] and say that there is an edge wherever f(x, y, t) == 0. The t dimension is shown vertically in slide 52. The way this approach allows continuously depicting multiple pieces is that as we slide the "mountain with two bumps" shape in slide 52 up and down (along the t axis) we get a transition from a single contour (around the base of the "mountain") to two contours (the "bumps"). With a more sophisticated family of functions we can have much more bumps, with more fancy shapes, allowing us to express many different contour shapes, consisting of different numbers of pieces. A way to understand the meaning of f(x, y, t) is that it should be positive on the inside of objects which we are "encircling" and negative outside - this way the zero-crossing will mark their edges, as we want it to be.