Superhero capes seem to come from two basic sources.

The first is The Shadow in particular and Pulp Era heroes in general. Many of these characters leaned strange abilities or unique fighting skills oversees and many of them adopted... not quite costumes, but outfits at least partially adopted from foreign lands or times. More than one wore a cape or cape like thing. The Shadow in particular seemed to have some sort of control over it causing his to move and billow. He regularly used it to cause his opponents to shoot his cape instead of himself. It was a well worn trope in the novels and radio shows, before "caped crusaders" appeared in comics. So of course a character like Batman ripped it off pretty early. Capes as fighting tools predate the comics and have been part of them ever since.

The second inspiration comes from circus performers. The cape was a common costume for trapeze artists and strongmen alike. And of course the stage magician with his cape was a common act in the 1920s and 30s. Depending on the act, the cape might be discarded before the act began, but it allowed a performer to make big flourishes, so they frequently stayed on. They were also useful for distraction, and also hiding. Nothing up my sleeve, but plenty hidden in the lining of my cape. Superman's initial costume is basically a strongman's outfit. A tight bodysuit, with trunks over the suit, and a cape over the shoulder. It would have been a well known sight to anyone who had seen a circus. In 1939, it didn't say superhero, it said strongman.

Drawing from the exotic outfits of the pulp mystery men and the (then) modern performers it's pretty easy to see where some of the early designs came from. And once Superman in particular and other early heroes in general become established, capes began to be common for later superhero comics. Once on the page a cape also allowed artists to use them as a sign of movement. A cape flowing behind a character could show speed and direction. Useful artistic shorthands tend to be preserved. And capes were certainly that.

I researched Superman for a college project.

Capes and cloaks were a popular clothing item up to the 18th century and thus many classic heroes such as swashbucklers and knights wore capes. For warriors and aristocrats in particular, capes were a means by which they could signal their allegiance or noble family, so they were designed to make an impression. Take for example this photo of Douglas Fairbanks in his costume for The Three Musketeers (Joe Shuster used Fairbanks' movies as a reference when he drew Superman). In the 20th century, pulp fiction artists used the cape to give their characters this heroic flair. Examples include the Shadow, Prince Valiant, and Mandrake the Magician, all of which predate Superman. Jerry Siegel and Joe Shuster were big geeks and really into classic adventure stories.

Something to note is that superheroes who wear capes tend to have been created during the 1930s and 1940s (the "Golden Age" of superhero comics). Superheroes that were created during the latter half of the 20th century tend to not wear capes (eg Spider-Man, Daredevil, Aquaman, etc.). Obviously by that time the influence of classic adventure yarns had waned.

Does anyone know if it also in part ties to the ancient "super heroes", such as Achilles, Beowulf etc.?

Capes were used in comics books to add a sense of motion (like when superman flys, his cape waves behind him), this would create an illusion of movement and could show the speed of a character's movements by where their cape is relative to where they are in the frame; in animation this is called a secondary action. Capes also gave superheroes and super-villains a "larger than life appearance", kind of like peacocking.