The answer to your question is multi-faceted, but I'll give you what I believe to be the most important reason.

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The Stetson company's Boss of the Plains style hat, which generated many styles and spawned many imitators, was the original hat design that became synonymous with the "cowboy hat." It was originally designed as a tall-crowned and wide-brimmed hat with a standard, slightly domed crown, but variations in the shape of the crown and brim were soon common. The most commonly accepted reason for these variations is that over time and with heavy use as the hat was picked up, put down, taken off, and put on the shape of the crown and brim were changed by the hands that touched it. If a man grabbed the front of the crown every time he took it on or off, the front of the crown became creased. If the man tipped his hat in the front, the brim might turn down slightly over time.

Certain specific crease patterns came to signify specific places, such as the Carlsbad crease featuring a top divot and two indented sides, taller at the back of the crown than the front. This style supposedly came from the Carlsbad region of New Mexico (hence the name), and cowboys from that region would intentionally reshape their new hats (or have the hatter shape them upon purchase) to show where they were from. Regional variations abounded, and over time hat manufacturers like Stetson and Resistol offered the more popular variations directly.

That is the commonly accepted answer to the question you asked, but I would like to offer some additional insight and context.

First, cowboys didn't wear the Stetson Boss of the Plains in large numbers in the years following the Civil War. Mexican vaqueros wore sombreros, former Civil War soldiers wore their kelpies, newly arrived cowpunchers donned derby hats, and people pretty much wore whatever they had. As much as the Stetson company likes to publicize that their hat won the west because cowboys everywhere immediately realized they needed a hat exactly like this, the truth is that cowboys wore Stetsons because life imitates art.

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Between 1869 and 1872, a few dime novel stories by author Ned Buntline became immensely popular with American readers, and these stories featured a pair of men who Buntline met at Fort McPherson, Nebraska, while on a temperance lecture. The men were working as civilian scouts for the Army and as a regular part of their job they were put in a position to guide Army troops across the prairies where they occasionally were involved in fights with Sioux warriors. One of those men was a former buffalo hunter named William F. Cody and the other a former cowboy named John B. Omohundro. Because both men were very adept hunters, they were also called upon to lead aristocrats like the Earl of Dunraven and Grand Duke Alexis on buffalo hunts, putting themselves in the national spotlight in newspaper stories printed in all of the big city newspapers along the east coast.

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So the exploits of these two men were being reported in newspapers and exaggerated in dime novels, and in late 1872 they joined author Buntline in a stage play, starring as themselves, that was incredibly successful. On stage the men used the names "Buffalo Bill" and "Texas Jack." The next year, Buntline left the touring group and was replaced with former lawman James Butler "Wild Bill" Hickok. Wild Bill soon decided that he didn't enjoy acting, but Cody and Omohundro toured together from 1872 until 1876, and then separately for another four years until Texas Jack's untimely death. Three years later, Buffalo Bill launched the outdoor "Wild West" show, becoming the most famous man on the planet and the most successful entertainer of his time.

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This is important because when the pair went on stage for the first time, they used the original Stetson Boss of the Plains hats as part of their costumes, along with fringed buckskins and elaborately beaded leathers. It is obvious in looking at newspaper reports that people weren't familiar with the Stetson hat, as reporters often talk about the men's sombreros and never mention them as cowboy hats or Stetsons at all. Though Texas Jack had been a real cowboy working the Chisholm Trail and introduced the lasso to the stage and wrote articles for magazines detailing his life as a cowboy, his hat was never called a "cowboy hat" during his lifetime. There are several advertisements for "Texas Jack hats" and "Buffalo Bill hats," but I haven't been able to find an advertisement for a Stetson "cowboy hat" until around 1885.

The reason this is significant where your question about brims is concerned is that early on, Buffalo Bill and Texas Jack wore unaltered Boss of the Plains style hats, but over time they both tended to alter the crowns and brims to allow audiences to see their faces while they were performing. This could take the form we associate with cowboy hats today, with upturns on both sides, or more exagurated turns depending on the need. In later pictures, they often are shown with highly upturned fronts for the sake of taking a good picture that still showed their famous faces.

People, cowboys and civilians, who wanted to imitate the most famous people wearing that style of hat intentionally copied their hat style, turning up edges and shaping crowns to look the part of these stage heroes. When Buffalo Bill's Wild West became America's favorite entertainment in the mid-1880s, that was when that style was advertised as the cowboy hat, as it was worn by cowboys in Buffalo Bill's shows, deliberately copying Texas Jack, Cody's former cowboy partner who had died just a few years earlier.

https://i.imgur.com/AnwKLyT.jpg
The image, taken in either late 1872 or early 1873, shows the cast of The Scouts of the Prairie. Buffalo Bill (second from left) and Texas Jack (far right) are wearing unaltered Boss of the Plains style hats.

https://i.imgur.com/vaFrGTE.jpg
Buffalo Bill Cody a year or two later, hat has a rolled brim, crown indention, and upturned right side brim.

https://i.imgur.com/vQqgjeX.jpg
Texas Jack around the same period. Though we can't see his hat crown, I assume it is indented much like Cody's, with a similarly upturned brim.

https://i.imgur.com/Ps8ZOJt.jpg
Later image of Buffalo Bill Cody. Eventually Stetson embraced the association and featured Buffalo Bill in advertisements. His hat heard featured upturned brims on both sides.