How much is known, or can be conjectured, about why the Greeks originally became interested in the somewhat arbitrary construction of intersecting a plane with a cone? The folklore that I've heard is that conics were forgotten until they turned out to be useful in describing the orbits of planets, making it seem even more unusual that the Greeks would have considered it worth writing entire books about them.
The truth is that we do not know. We do know of the person who is credited with the discovery, Menaechmus (c. 350 BC), a student of Eudoxus of Cnidus and a friend of Plato's, one of the most prominent mathematicians of his time. The names ellipse, parabola and hyperbola were given to them by Apollonius of Perga over a century later however. Menaechmus called them literally acute section, right section and obtuse section because they can be obtained by sectioning cones with the corresponding apex angles perpendicularly to the generator. Others also called them Menaechmian triads.
The traditional answer is that the reason was the Delian problem, the most famous of the "three construction problems of antiquity". The oracle of Delphi allegedly told the citizens of Delos to double the size of a cube shaped altar to stop a plague, and when they doubled the sides and nothing happened, specified that it was the volume that was to be doubled. The part about the plague in Greece at about the right time can be confirmed, as can be the Greek habit of consulting the Delphic oracle. The rest is neither here nor there.
Taking the side of the original cube as a unit doubling of the cube reduces in modern algebraic notation to solving $x^3=2$. Greeks had no algebraic notation, and their favored method of solving geometric problems was using straightedge and compass. As we know today such an $x$ can not be constructed with those tools. Hippocrates of Chios noticed however that one could find such $x$ by solving the double proportion $1:x=x:y=y:2$, or as Greeks put it, "inserting two mean proportionals" between $1$ and $2$.
It is a trivial manipulation today to see that Hippocrates's proportion is equivalent to a pair of equations $x^2=y$ and $y^2=2x$, describing two parabolas, or $x^2=y$ and $xy=2$, describing parabola and hyperbola. If they can be constructed then the intersection point will give the solution to the Delian problem. Menaechmus's task was considerably harder. He could not manipulate coordinate formulas, Greeks only had a crude prototype of them called "symptoms", nor did he presumably know of curves with such symptoms ahead of time. So he had to reverse engineer symptoms from Hippocrates's proportion, and then have an insight that curves with exactly these symptoms can be obtained by sectioning cones.
If this seems hard to believe, it is. It is possible that Menaechmus got some clues from previous, mechanical, solutions to the Delian problem by his teacher Eudoxus, and by his teacher's teacher Archytas of Tarentum. It is also possible that he experimented with sections of the cone for other reasons, ellipses implicitly appear in Eudoxus's homocentric astronomical models, for example, and noticed that they have the requisite properties to solve the Delian problem. See Yavetz's A New Role for the Hippopede of Eudoxus and Riddel's Eudoxan Mathematics and the Eudoxan Spheres for geometric details of such speculations.
This question has been discussed several times on math overflow: https://mathoverflow.net/questions/191909/discovery-and-study-of-conic-sections-in-ancient-greece
It also has references.
One theory is that they appeared when the Greeks started to think how to make accurate sundial. This theory is developed in several books and articles on the subject, and the top scoring answer in the MO question I cite above is about this theory. However on my opinion, shared by some historians of mathematics, this theory is not sufficiently justified.
A more plausible theory is that they were discovered when attempting to double the cube.
That conic sections were forgotten with the decline of Hellenistic mathematics is correct. But this applies also to the other great works of that epoch (Archimedes). This followed by a complete collapse of science, and for about a millennium there were simply no people who were able to understand Apollonius or Archimedes. Then Kepler discovered that planets move on conic sections, and a bit later Pascal proved first new theorems about them.
I beg to differ with Conifold's answer, and say that in this case the truth is more likely, that we do know why, reasonably well. There are at least two strong practical reasons to study conics, besides the mathematical interest per se, both from physics, precisely optics and acoustics.
We do know that the famous Lighthouse of Alexandria had a parabolic projector, which was visible at a distance almost 50 km, according to Iosephus. Knowing the focal properties of the parabola, discovered by the Greek geometers, are necessary in order to build such an advanced technological device, which was so admirable to be named "the seventh wonder of the world".
We know that theaters were extremely important in the culture and social life of the Greek world, and we do know that they were carefully projected and built, in particular, in order that the voice of the actors be clearly audible by the (often more than 10,000) spectators. Again, the knowledge of the focal properties of conics, that we do know they did know, are mandatory to this aim. The acoustic effect is still observable in the remaining Greek theaters, where the conic shapes and the location of the focus can be measured.
For a detailed account of the Hellenistic science I recommend Lucio Russo The Forgotten Revolution.