When Carmen premiered, neither critics nor audiences enjoyed the opera. What changed about attitudes towards opera that it has become one of the most well-known pieces in the world?
I've mused on this for a while, because this is an OPINION question and those are really hard to answer in their own way! You have to chose from an infinite number of answers and argue your way into one and out of the others... But here's four things I think should be considered:
Paris Opera was (probably still is) a very conservative opera house, a state-sponsored opera house, and almost a family opera house. This opera was too sexy for them. You take your teen daughters to what you think is going to be a Disney movie and you see an unrepentant adulteress get murdered on stage, with tax payer money? You'd be cheesed too. Outside of the Moral Majority, Parisian cognoscenti were also pretty anti-Wagner, and Carmen was thought to be too threatening to traditional French music and Wagnerian-influenced. So it had two problems, it probably would have done better in France if it only had one.
It wasn't truly a failure at its premiere, it certainly wasn't a "critical failure," some critical people appreciated it, it was more of a will-it-play-in-Peoria failure. It had 48 performances for its original run in Paris, so they didn't close it early (the mark of a true flop opera), and people came to see it even after word spread that it was trashy, precisely because it was trashy. It's just harder to have a "cult classic" develop in an artform as expensive as opera, you can't do like RHPS midnight runs of opera in 19th century France, so it wasn't performed again in Paris for years.
When it got revived to a less conservative opera crowd (Vienna) 6 months after its premiere in Paris it was a hit. The Viennese, who weren't anti-Wagner anyway, didn't interpret any of the music as Wagnerian, and thought it was just a new French opera, so no problems there. The amoral plot also, well that was just the French you know, and accepting an amoral art from a different culture is a lot different than accepting it from your own. The fact that it got revived at all means contemporary people recognized it was a good piece, and wanted to give it another shot. It's not like some of the true operas that have been rejected and rescued, like Handel's Serse, which wasn't performed for about 200 years. People in the opera's natural lifetime recognized it as good.
It was edited when it was revived! One element which was found distasteful then, spoken dialogue, was replaced with composed recitative. People weren't so stuck up about editing operas back then. I think the removal of one artistically risky move helped people outside of Paris accept it better, although that's up for argument. But the version that the world learned to love was this edited version of Carmen, although the version we see now is more "authentic" and has the spoken dialogue, which opera crowds don't even blink at today.
So, a combination of the composer, librettists, and impresario all not really giving a poop about what the Paris Opera box holders wanted to see is the only reason it got a reputation, and its reputation was rescued in about 6 months. So not so bad!
This is one of the standard reference books on Carmen which I consulted today, but there is an overview in New Grove Dictionary of Opera that I think is a good summary of its complicated reception. The Wikipedia article is also pretty okay!