The glass slippers can be traced to exactly 1697, in the version by Charles Perrault. All versions which mention specifically glass slippers are derived from this version. Others have the slippers being valuable or unique in some way, but never glass. Gold was common. Other versions do not mention the slippers themselves as important: they're simply a mechanism in the story.

Glass slippers (as in shoes made entirely of glass) have never existed. In the original story, they are included in a list of other things that have never existed like lizard footmen and pumpkin carriages. They are important because they are the one thing the Fairy Godmother gives instead of transforming something else, and thus contain a mystical quality to them.

Indeed, they are the only thing Cinderella gets to keep of the night. The fact they cannot exist is part of the point. Their uniqueness also validates the test at the end: they would be completely rigid (and so you could tell if they fit exactly) and they would be unique, so no one else would have a matching slipper around but their original owner.

Glass would have read to a contemporary audience as fragile and valuable. There had been a vast expansion in glass production so it was filtering down to the gentry and prosperous merchants as a status symbol by the end of the 17th century. Indeed, the price declined sharply from about the 16th to late 17th century. But it was still valuable: the pair of glass slippers (if real) probably would have cost about a month's wages for the average laborer. And that's presuming they were the same quality as "normal" glassware and not something more elaborate. Another way to think of it: they'd have been about half as expensive if made from silver. Another another way to think of it: the price of two nice pieces of glassware could have paid a working class Parisian's rent for nine weeks.

Notably, it would have also read as temporary. 17th century glass degraded over time into so called "sick glass", sometimes very quickly in the case of either poor quality or especially clear glass. Slippers, meanwhile, would have been taken as a reference to court shoes.

Further, it's important to keep in mind Perrault was writing morals. The primary moral of the story (and this is not guessing, Perrault tells us what the moral is) is that it's important, above all else, for a woman to be graceful and gracious. By literally dancing in glass slippers, Cinderella is proving herself impossibly graceful: she can somehow dance on glass slippers without breaking them. This is possibly meant to be magical but regardless is meant to emphasize the point that her gracefulness was important to her success.

(The secondary moral is that sometimes, no matter how beautiful or gracious you are, you need a little help from a fairy godmother. Interestingly, it goes out of its way to point out that beauty, while good, is less important than being graceful of form, gracious of attitude, and having people who help you.)

But yes, more broadly: a wizard fairy godmother did it.

While there is some very good discussion here from another, a few additional things are worth pointing out. "Cinderella" is one of the most widespread of the European folktale types, and it has a staggering number of recorded variants. It is classified as ATU 510, a designation that typically implies historical diffusion, but this is difficult to accept in all cases given the widespread nature of the variants. Anna Birgitta Rooth (1919-2000), one of many excellent students of my mentor's mentor, Carl Wilhelm von Sydow (1878-1952) wrote the classic study of the Cinderella cycle of folktales for her doctoral theses (1951). All studies of the cycle must begin with her work.

As indicated elsewhere, the motif of the glass slipper is an outlier. It appears possible that it was the invention of the French collector and publisher of folktales, Charles Perrault (his collection appearing in 1697), but we cannot be certain whether or not he did, in fact, invent this most memorable aspect of the Cinderella story. It is possible that he heard it told this way, but simply cannot know.

To your question about it actually existing, etc. - and behind that question, if such a thing did not exist, how was this motif accepted by an early modern European audience (or an early folk audience for that matter!): to answer this, we must understand that the folktale was a genre of oral narrative told as fiction. These were the adult novels of the folk, and a folktale might take several nights in the telling. Audiences expected these stories to delve into the fantastic - indeed they wanted that. Folktales contrast with the legend, an oral narrative generally told to be believed. Legends clung closely to the realistic and believable, typically introducing a single extraordinary (if not supernatural) motif in a believable setting so the whole ball of wax could be accepted, but folktales were by rule extraordinary - and fantastic in the fullest sense of fantasy.

I spent a year (long ago in a previous century!) studying all the variants of ATU 306, the Twelve Dancing Princesses. In this folktale, the princesses, and finally the man who would break their enchantment, travel underground through forests with leaves made of precious gems. This is a place of a magical castle surrounded by magical forests. The audience accepted this - just as they still do with the glass slipper and the setting of ATU 306 - because the audience expected the fantastic extravagance. The audience of the genre understood - and understands - that they were/are in for a wild ride. Believablity was not the point. Glass slippers didn't exist, but neither did forests with trees filled with gems.

Let me know if you need more!

Additional question: Where did the "glass slipper" start? I had always considered it a Disney invention, that then hung around in the anglosphere.

The original text, Kinder- und Hausmärchen, 1850 talks about "mit Seide und Silber ausgestickte Pantoffeln." = "with silk and silver embroidered slippers" and it stayed that way in the story books of my childhood (german language).