Why did Melisandre believe that Stannis is the reborn Azor Ahai?

[three-eyed monkey]

The answer to the question posed in the OP is quite simple. Mel believed Stannis was Azor Ahai because she saw him standing before the dark with Lightbringer raised. She then shoehorned Stannis into the prophecy by claiming that born amidst salt and smoke, etc., fit with Stannis, in a clear case of confirmation bias. She then goes on to put "Lightbringer" in his hand so that he fits all the more, and as such this is becoming a sort of self-fulfilling prophecy on Mel's behalf.

Mel claims she is one of the best in her order at reading the flames, but we know she is pretty sketchy. She gets it right at times, for example she seems to have foreseen the deaths of Balon, Joffrey, and Robb, information she cleverly exploited using some leeches. She makes mistakes too, and often only gets things half right-ish. She sees a girl fleeing a wedding on a dying horse but it's not Arya. She asks for a glimpse of Azor Ahai, who she believes is Stannis, but all she sees is Snow. She does seem to be able to extract some truth from the flames, and gets accurate glimpses of the future, but often misinterprets what she sees. She often gets some details right, but other parts of what she sees seems quite vague.

So what did she get right and what did she get wrong about her vision with Stannis and Lightbringer? I believe she did see Stannis with the sword she gave him raised against the dark. But she is misinterpreting the dark. She believes it's the Long Night, or the Great Other, but in fact the dark is a shadow. Personally, I think it's quite poetic storytelling.

I base this on the fact that I believe it is clear from Stannis' story that he will ultimately destroy himself, or perhaps more accurately be consumed by his dark-side or shadow, which was unleashed in his Faustian pact with Mel and her sorcery. The more adversity he faces on his quest for the throne, the more his dark-side emerges, the more people he burns, and the more of his humanity he loses. Shireen represents the last shred of Stannis' humanity, and when he burns her in a misguided attempt to attain his goal, the last of his humanity will go up with her and his shadow will have consumed him entirely. Of course, this won't kill him.

Stannis will burn Shireen to wake dragons from stone. The reason he will need dragons is because he will be facing Dany, who has dragons. She will stand between Stannis and his goal, the Iron Throne, given that it is her goal too. The confrontation between Dany and Stannis was set up in her vision at the HotU, where he was the stone beast who took flight from the smoking towers of Dragonstone, breathing shadowflame. I suspect Dany will confront all the kings in her vision, such as the mummer's dragon and the rest, one of whom she has already met.

The problem for Stannis is that he draws his definition of true sacrifice from Mel, who misinterprets the Azor Ahai legend. The murder of Shireen will not be a true sacrifice, because the only true sacrifice is self-sacrifice, and therefore it will not work. No dragons from stone for Stannis, meaning Dany will be free to swoop in on Drogon and melt his crown, while he can do nothing more than wave his fake sword at her, the one Mel put in his hand. And if Mel had been a little more accurate in her vision she would have seen that, yes, it was Stannis with his sword, but the dark is not the Long Night. It was Drogon's shadow.

At least, that's my take.