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There's no clear answer. Possible Reasons:

1. "Bond, James Dragon Bond" (Strength: Strong)

Dragons and their riders share a deep bond and Dragons come to share the likes, dislikes, enemies and friends of their riders. For example, Rhaenyra Targaryen's sons could go and sit on her dragon (But couldn't take her to the skies) but whenever Rhaenyra's dragon came anywhere near Aegon II's dragon, they'd both snap and roar at each other and spit flames. Same was the case for all the members of Green and Black parties. So if Daenerys to the last moment thought Jon a friend, Drogon would think likewise.

2. HAHA, It was me all along! (Strength: Medium)

If the bond between Dragons and Dragonriders is similar to the bond between a warg and his beast, it is possible that some part of Daenerys went on to live in Drogon (Since we know from Varamyr Six skins that a warg ends up in their beast once they die - And we know Bran felt dead wargs inside the crows in 3ER's cave) and it was that part which made the decision to spare Jon, melt the throne and leave forever. But that wouldn't hold since Drogon wasn't sure what happened and kept trying to wake her up, surely if Dany was within he'd know. But of course the counter argument is that Dany was not entirely in control of him, she was just getting accustomed to the new body in the beginning. The real control begins with Drogon's shriek which would be similar to Hodor's rage whenever Bran tried to slip into his skin.

3. "You hurt mommy! I am not gonna call you dad" (Strength: Weak)

Drogon realised with Daenerys' death, it was now "Finis 'Pax' Targaryenorum". Jon might be a Targaryen but his hands are tainted with blood of a Queen, the last unapologetic Targaryen which makes him a Queenslayer and a Kinslayer who betrayed his family and his family's words, which in turn makes him unworthy to ever sit the throne that his namesake and ancestor Aegon the Conqueror forged with Fire and Blood. Of course Targaryens have been killing other Targaryens for ages yet the Dragons keep serving so it's not for Drogon to judge and sentence Jon. But he thinks Jon may have done it for the throne. What should he do? Kill the last Dragonlord? Serve the last Dragonlord? Leave but destroy the throne so that kinslayer, traitor could never sit on it? He chooses the middle route, spares Jon, destroys the last visage of Targaryen legacy in Westeros so that people unworthy of it could no longer sit in it and flies away. Verrrrrrry Weak speculation.

4. "Hate the game, not the player" (Strength: Medium)

Drogon blamed the Iron Throne for deaths of his mother and brothers. He didn't blame Jon, he blamed the throne for taking them away from him and he completed his mother's mission by melting it. The Quest for throne after all made Daenerys what she became, killed Rhaegel and Viserion and eventually killed her too. It doesn't matter whose knife it was. Daenerys became the Jenny in the song Jenny of Oldstones. She was dancing alone with her ghosts in the halls of Kings who are gone, The ones she'd lost (Aerys, Rhaella, Rhaegar, Aegon, Rhaenys), the ones she'd found (Viserys, Rhaego, Drogo, that child Drogon ate), the ones that loved her the most (Jorah Mormont, Barristan Selmy, Missandei). Drogon is thankful that Jon sent her to a better place where she can dance free of her pain, Jon has in fact spun away her sorrow and pain with one swift prick.

5. "Thou shalt not kill thy kin" (Strength: Weak)

Drogon felt the affinity that Blood of the Dragon feel for each other when he came across Jon. But that is particularly weak since Dragons can and have killed Valyrians who they perceive as threat to their riders. And if they recently became riderless, they are particularly bad tempered. For example Prince Aemond recalls when he claimed Vhagar on funeral of her last rider Laena Velaryon. He knew his parents would never allow him to try and claim the largest dragon in the world at that time due to grave danger, he sneaked out to the dragon's pen early in the morning. Aemond says he was so afraid of getting caught and scolded by his father that he forgot about being afraid of getting eaten by the Dragon.

6. "All my friends are dead" (Strength: Strong)

Dragons can actually get depressed and go mad with grief, even though they are creatures of Fire and Blood, if they lose a loved one. The Dragon Silverwing is said to have gone mad with grief when her long time mate Vermithor died. She is said to have tried to make Vermithor rise up like Drogon did with Daenerys and after she failed, she took wing, never to return to King's Landing or Dragonstone, staying out of the wars and living out her days peacefully in Redlake. It is quite possible that something like that happened with Drogon, he bore the wounds of losing his siblings but losing his mother finally cracked him. He now seeks peace and tranquility with his own ghosts to keep him court, he's done with violence.

I want to add a different answer. You mention that Drogon "seemed to understand what happened", but I'm not so sure. Dragons are intelligent yes, but I do not believe a dragon has the capacity for investigative thinking that would allow him to deduce that Jon killed her. She was dead and Jon seemed respectful of the body. It is possible that Drogon saw Jon as just another person mourning the death. I'm open to be proved wrong if someone has evidence that Dragon's can deduce this well. Recognizing death is one thing, but thinking "you killed her!" is another.

Because Jon Snow is in reality a Targaryen too. And not just any Targaryen, but Aegon Targaryen, the rightful heir to the Iron Throne for that matter. Therefore the dragons obey him as well, and would not harm him. Same reason they let him ride them.

I have other theories :

• he thought Jon was right to kill her because he thought himself she was gone mad


• he expected Jon to be punished by unsullied anyway, in a more painfull way than being burnt alive (because from Grey Worm's previous behaviour toward Jon, this would definitely have been my personal expectations)


• he knew that if he killed Jon, the entire North would chase him to death

The reason he destroyed the Iron Throne would be :

• He wanted to prevent anyone to go mad like Dany


• He wanted to send the message : if she doesn't go on the Iron Throne, nobody will, maybe interpreting the murder of Dany as a prideful way to take the throne from her

Or because Jon didn't moved much, he didn't see him, because his vision is based on movement like T-Rex in Jurassic Park.

Or maybe scenarists had enough of GoT at the end and made a smooth/mawkish final because they didn't care anymore anyway.

Or maybe they didn't have any model or skeletons for burnt corpses as they were all used in the previous episode.

Dragons are smart. Could he see Snow's fingerprints on the weapon? Probably not. However, I think he nuked the Iron Throne instead of Jon Snow because it wasn't Jon Snow who truly killed her. Drogon knows her desire for the Iron Throne, and that it drove her to a point that ended up killing her. She wasn't the same Danaerys at that point, and his "mother" is now gone.

I believe this is what the creators wanted to show. He still loves his "mother", but could recognize the monster the Throne had driven her to become. So he destroys the evil, takes her body and leaves with his sorrow.