This article contains heavy spoilers. Proceed with caution
Even though a lot of us felt the 'Game of Thrones' finale didn't deliver apt justice to what we actually expected, it was definitely more closely-knit than most episodes in the final season. Mostly because it was the last episode and the writers tried their best to tie up as many loose ends as possible!
Daenerys' death at the hands of her lover-cum-nephew Jon Snow, although tragic, was already set in motion since the entire Azhor Ahai prophecy was talked about by the Red Woman aka Melissandre, earlier in the show. 'The Prince That Was Promised' did show up in the end and kill the love of his life, but more about that later.
For now, let's focus on the Mother of Dragons' only surviving son Drogon, who saw his dead mother and made it visibly obvious that he was mighty upset.
After a conversation with Tyrion, who spoke against Dany being a fit queen, Jon went to find her. While entering the castle, Drogon stood in front of him, making sure no one else goes inside, being protective about his mother. When Jon got to Dany, she was standing near the throne, and we all knew what was going to happen.
The only time Jon would kiss his aunt was when he'd wish her a final goodbye. And he did. He kissed her and plunged a knife into her heart. Surely enough, Drogon got a sense of what was going on and saw his mother's lifeless body lying on the floor.
Interestingly enough, when he (Drogon) found out his mother was dead, he didn't turn towards Jon and roast him on the spot. Instead, he turned towards the Iron Throne, the cause of everything, and melted it down with his fiery wrath.
Drogon did that because he always knew of Dany's obsession for acquiring the throne. And he also knew that her obsession got the better of her, when she burnt down thousands of men, women and children in order to sit on it. So, for Drogon, Jon was never her murderer but the Throne, which is why he let Jon live.
Having said that, Jon also happens to be a Targaryen and Drogon definitely recognised that, especially in Season 7, when he let Jon touch him and season 8, when Rhaegal let Jon ride him. Only a true Targaryen can probably tame a dragon and hence, a dragon would never burn down a Targaryen, with the simple rule of blood and fire.
Now, coming to the matter of where Drogon took Dany's body and flew away. Remember in the later part of the episode, when the first of its kind Small Council Meeting takes place? Bran asks about the whereabouts of Drogon and it's revealed that he was last seen flying East from King's Landing. Well, the only probable place East of King's Landing is Valyria and that's probably where he's taken Dany too.
Valyria is an old Targaryen dynasty, which was home to all the Targaryens and their dragons. Also, when Drogon went missing in Season 5, he was spotted by Tyrion and Jorah, when they were passing through Valyria. Of course, Drogon knew where home was and he took Dany there in all probability to lay her body.
Also, in the Season one finale, before Dany acquires her many titles, she originally referred to herself as "Daenerys Stormborn, of the House Targaryen, of the blood of Old Valyria." She specifically names Old Valyria knowing that's where her family and her ancestors are from and that's her real home, which gave her end, when Drogon flies her away, probably back to Valyria, an absolute poetic justice!