Short answer: fashion changed.

Longer answer:

What is a corset? It is a structured undergarment with 2 main and 1 supplementary purpose: support the bust and back; create the currently fashionable silhouette; provide weight distribution to heavy skirts and dresses. The corset itself is the 19th century iteration of those 2.5 purposes.

In the late middle ages the stiffened kirtle bodice came into fashion. A kirtle (underdress) would have the bodice boned or otherwise stiffened to provide bust support and create the at-the-time popular smooth front look. The stiffened bodice stayed popular for hundreds of years and was worn all through European history till the turn of the 19th century when skirts rose right up to the bust. At the same time a separate garment came to exist. First called a pair of bodies, then stays, then finally a corset, this was an item that supported the bust and created the contemporary attractive upper body silhouette. It's important to remember that these are not the same garment. I know less about bodies but stays were used to literally pin your clothing to you as many of the popular styles of the 17th and 18th century consisted of open robes (dresses) with a stomacher (decorative front panel).

Corsets came about in the standard way, they were developed out of a need for new structural garments to fit the contemporary fashion. In the late 1700s waistlines softened and started to rise. This eventually made older stays obsolete and new short or long stays (and even bra-like corded wrap things) came about to support and lift the bust for the popular "resting on a shelf" look. Once the waistline dropped in the 30s longer corded stays/corsets developed, now with a defined waist. As fashion changed, so did the corset and by the 1850s we have the corset that everyone thinks of: split metal busk, cross lacing on the back, defined waist, metal boning, lack of shoulder straps.

Fashion in the 19th century changed rapidly and in just a few decades this style of corset became obsolete, because fashions now emphasized a smoother front and a more visible hip while leaving the bust loose. The corset moved down over the hips (just the fabric portion, not the boning) and eventually came off of the bust. And then WW1 and the 1920 happened.

The corset didn't disappear but it did become a lot less popular and mostly worn by older people who were used to them but clearly the very loose and almost rectangular silhouette of the 20s didn't need a corset, Did need a bra though, and bras have been our bust support standard ever since. BUT. But. The need for a structured support and silhouette creating garment didn't disappear, it just changed. From the girdles of the 40s and 50s to the spanx of today we still use our equivalent of the corset, we just split it into separate pieces for bust support and shaping and stomach and hip shaping.