Why did Dany decide to torch King's Landing after the bells were rung?

Question

In S08E05 of Game of Thrones, Dany decides to torch Kings Landing and all the innocent people in it. She is angry and can't seem to hold back her rage, and apparently taken with the madness that afflicts Targaryens.

But why does she attack the city and the people first?

Why not go after the cause of all her woes, Cersei, who's right there in front of her, whilst Dany looks directly at the Red Keep where Cersei is standing.

Dany knows that Cersei doesn't care about the inhabitants of King's Landing. She can't be killing them just to cause Cersei pain. And in doing so she gives Cersei plenty of time to escape, missing her best opportunity.

Answer 1

Dany has finally crossed-over to the "Mad" side

There has been a lot of subtext and foreshadowing pointing towards Dany being just like her father; this is merely the final straw that makes her go Full-Mad (never go Full-Mad).

Targaryen lineage and a history of Madness

You'll recall the line that Ser Barristan recounts about Targaryen disposition:

King Jaehaerys once told me that madness and greatness are two sides of the same coin. Every time a new Targaryen is born, he said, the gods toss the coin in the air and the world holds its breath to see how it will land.
^{-A Song of Ice and Fire: Book Three - A Storm of Swords (Chp 71, Dany VI).}

And recalled by Varys in the recent episode of Game of Thrones, Season 8 Episode 5 (The Bells), as well as in the 'previously on' section.

This has been setting up the potential for Dany to go Full-Mad at any point.

A lead-up

This is not the first time that Dany has chosen such a rash decision, burning people alive, and sometimes when she has the chance to show them mercy:

• In Season 7, Dany burns Randyll and Dickon Tarly, after they have been captured as prisoners of War^[1]
• In Season 3, Dany burns Kraznys Mo Nakloz, in an attempt to free the slaves^[2]
• In Season 6, Dany burns the Temple of the Dosh Khaleen, in an attempt to escape her captivity at the hands of Khal Moro^[3]
• In Season 1, Dany burns Mirri Maz Dur in Drogo's funeral pyre, in an attempt to seek revenge on for her husband and to awaken her dragons^[4]

This has always been very very reminiscent of how her father would deal with things

Burn them all! BURN THEM ALL!
^{-Aerys II Targaryen, Game of Thrones.}

Even in the books, there has been a lot of suspicion about her madness, see: Targaryen Madness.

Dracarys

So this brings us to the final point, Dracarys and Daenerys' familiarity with this move. In a strange parallel, Daenerys uses this move as comfortably as Harry Potter uses Expelliarmus; often and liberally.

Coupled with the fact that it was Missandei's last words^[5] and how it really tore Dany up. She wasn't eating, had locked herself away in her room and literally only came out when she heard that people were conspiring to take her Throne away from her. She was mad, like, incredibly angry, upset, frustrated and felt alone, with no love and surrounded by fear.

This final bit of betrayal is the culmination of a prophecy made about Dany, in the House of the Undying:

mother of dragons ... child of three ...three fires must you light... one for life and one for death and one to love... three mounts must you ride... one to bed and one to dread and one to love... three treasons will you know... once for blood and once for gold and once for love...
^{-Dreams and Prophecies.}

So she finally realises that she is alone, has been betrayed and will have no love, whether she takes over peacefully or not.

So she decides to:

Burn them all! BURN THEM ALL!
^{-Aerys II Targaryen, Game of Thrones.}

Answer 2

^{I answered a similar question on movie.SE, so I just copy the answer here}

She has become mad pragmatic

While some will argue she has become as mad as her father, I will argue she has all her mind.

One important element to understand this is the dialogue she had with Jon Snow before the attack

Far more people in Westeros love you than love me. I don't have love here. I only have fear [...] Alright then," she says. "Let it be fear.

She knows she won't earn love from his people. And as Machiavelli says:

From this arises the question whether it is better to be loved rather than feared, or feared rather than loved. It might perhaps be answered that we should wish to be both: but since love and fear can hardly exist together, if we must choose between them, it is far safer to be feared than loved
^{the prince (1513)}

As Benioff in the "inside the episode":

I think that when she says "let it be fear" she's resigning herself to the fact that she may have to get things done in a way that isn't pleasant and she may have to get things done in a way that is horrible to lots of people.

Why did she react like this to the bells?

However, she is not totally emotionless, and it's true that there is rage inside her when the bells tolls. Because at this moment, she realised that she could have easily take Kings Landing, two seasons ago.

She realized she could have dealt with the Night King threat after becoming Queen.

She lost two dragons, Jorah, Missandei, and a big part of her army, while a simple attack with only one dragon was enough.

She realized all those previous sacrifices were useless.

How can she be such a good ruler in Essos, but so cruel in Westeros?

While it seems odd that the same character rule in a quite different way between two realms, this is not unrealistic. Bruce Bueno de Mesquita, political scientist and game theorist, studied the reign of Leopold II, King of Belgium and Congo. The exact same person rules these two kingdoms in very different ways, but there is a reason for this: the power was not held by the same people/structure. Two different kingdoms, two different ways to conquer/keep the power.

Abstract
From 1885 until 1908 Leopold II was not only the King of Belgium but also the personal owner of the Congo Free State. The policy outcomes during his reign turned out to be fundamentally different in the two countries: Whereas in Belgium he improved living conditions, in the Congo he established a brutal tyranny. This paper analyses the reasons for these different leadership styles of Leopold II by means of the 'selectorate theory'. The selectorate theory explains policy outcomes as a function of governance institutions. It assumes that the ruler maximizes his own utility which means first of all to sustain himself in power. Under Belgium's governmental institutions Leopold II required broad support from the general public but in the Congo he only needed a very small group of supporters. To reduce the possibility that Leopold's different leadership styles were caused mainly by racism his period is compared to the reign of the Congolese leader Mobutu Sese Seko.
^{Leopold II and the Selectorate: An Account in Contrast to a Racial Explanation, Bruce Bueno de Mesquita}

The same thing can be said for Essos and Westeros. In Essos, she is seen as a liberator and can earn the love of her people, whereas she is seen as a conqueror in Westeros, and must rule with fear

Answer 3

Other than what Mooz has mentioned, D&D, the minds behind the last episode put their reasoning in this way (Relevant parts only):

Benioff: Dany is an incredibly strong person. She's also someone who's had really close friendships and close advisors for her entire run of the show. You look at those people who have been closest to her for such a long time and almost all of them have either turned on her or died, and she's very much alone. And that's a dangerous thing for someone who's got so much much to feel that isolated. So at the very time when she needs guidance, and those kind of close friendships and advice the most, everyone's gone.

[...]

Benioff: Jon Snow is someone she's fallen in love with and as far as she is concerned, by this point, Jon has betrayed her by telling people about his true identity and also the fact that he's unable to return her affections at this point.

Weiss: I think that when she says "Let it be fear", she is resigning herself to the fact that she may have to get things done in a way that is not pleasant. And she may have to get things done in a way that is horrible for a lot of people.

Benioff: She chose Violence. A Targaryen choosing violence is a pretty terrifying thing. Even when you look back to Season 1 when Khal Drogo gives the golden crown to Viserys, and her reaction of watching her brother's head melted off......And he was a terrible brother you know so I don't think anyone out there was crying when Viserys died but there's something kind of chilling about the way that Dany has responded to her enemies deaths. And if circumstances had been different, I don't think this side of Dany ever would've come out. If Cersei hadn't betrayed her, If Cersei hadn't executed Missandei, If Jon hadn't told her the truth, Like if all these things had happened in any different way, Then I don't we'd be seeing this side of Daenerys Targaryen.

Weiss: I don't think she'd decided ahead of time that she was going to do what she did. And then she sees the Red Keep (Warning: Flash imagery), which is to her, the home that her family built when they first came over to this country 300 years ago. It's in that moment, on the Walls of King's Landing, where she's looking at this symbol of everything that was taken from her, when she makes the decision to make it personal.

So basically:

1. She is very very lonely and she has no friends left that she can rely on.
2. Everyone she relied on either betrayed her or died in her service, which did no favours to her mental health (Apparently).
3. Hell hath no fury like a woman scorned.
4. She decided that she will get things no matter how dirty her hands get. She's sick and tired of indecisiveness. She was willing to at least try and be gracious even if she feels dark inside but when Jon refused to answer her when she asked "Your queen, is that all I am to you? and didn't return her kiss, she decided oh well, might as well go dark publicly.
5. SHE WAS ALWAYS MADDDDD! The reaction to her brother's death (The man who abused her all her life, tried to rape her, sold her like a horse and before his death threatened to cut out her unborn son) and her enemies (Those poor slavers who crucified babies) is veryyyy terrifying.
6. Cersei meant so much to her for some reason that Cersei's betrayal was the one that she could not take, that was the one that pushed her over the brink. Maybe she loved Cersei? Who knows. We might find out in the next Inside the Episode.
7. After seeing that symbol of everything that was taken from her family, the Red Keep (Nevermind the fact that she'd just retaken it), she decided to unleash fire and blood, because why the hell not? That'll show them.

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