One of the main reasons on why I love Dany is because of the journey she took to achieve her goals. She started off as a shy girl with low self-esteem who was mentally and physically abused by her own brother and then she was sold off like a piece of meat to Kal Drogo, she was a slave. At the moment she had two choices, accept the current reality, just go with the flow or stand up for herself and make the best out of the situation she was in, and she choose the latter.
Slowly, she started to realize that she had more power and potencial than she ever imagined and because of that, she started conquering slave cities and freeing the slaves, because she could. And that's the thing, she wants to the right thing, she doesn't want anybody to go through slavery anymore, she wants to eradicate the horrible culture dominating these cities, like she said to the Greyjoys: "We're going to leave the world better than we found it."
With that ideology, she wants to change Wasteros, she knows how horrible things have been in there, specially how terrible the lords of Westeros have treated the common folks and how the power has corrupted the goverment in King's Landing, something that even Robert Baratheon noticed. All the time she speaks about taking the throne and how it belongs to her and sure, that can be annoying, but people forget that she just doesn't want to sit on it and enjoy the life of a Queen just because, no, she actually wants to completely change how the system works and make the system better for everyone and not just for the ones at the top.
And if she is Queen, she will break that wheel since she can't have a child and thus, I think she will create a system where the throne is earned and not handed down to your children, but that's just my speculation, I just wanted to point out that Daenerys wants to change the world for the good. So yes, just like Jorah advised her to, she needs to show strength to her people and that's why she needs to let people know that she is the rightful ruler of the seven kingdoms. From time and time again she gets underestimated, insulted and mocked and yet, she proves them wrong, they tell her you can't do this and that, because things are like this and that and she ends up doing the imposible.
Of course, I have failed to mentioned her biggest flaws, like her hunger for power, her being a poor ruler and her idea of what "justice" is, although she is slowly learning and the most valuable thing that she has learned from Meeren is that she needs to listen to her counselors, even Tywin said to Tommen that in order to be a good king, you need to listen to your counselors.
In the end, people like Varys and Tyrion said they believe in her and Dany will face many challenges in this coming season and most of those challenges will be overcoming her weaknesses. The point that I wanted to make with this wall of text is that Daenerys is also a complex character, just like the rest of the characters in GOT and I think it's unfair when people only focus on her flaws and not her strengths. But that's understandable, she's the only character who after a lot of struggle, gets a victory at the end, unlike other characters who put a lot of effort into things and end up even worse and I can see how that feels unfair.
So yeah, while others see lines like "I was born to rule the seven kingdoms and I will" as annoying and childish ( And I can understand why a person would see it like that), I see it as badass. Plus, out of all the characters, I feel more related to her and her story.