A number of complex reasons, actually.
First off, you've got her sociopath of a father dumping all his expectations on her. Everything she does has to be perfect. He's shaping her to be a tiny version of himself -yet at the same time, it's obvious he thinks of her as disposable. A weapon to be used and discarded. Ozai fed her all his twisted views; that no one can love you lest they betray you, that loyalty can only be bought through fear, that manipulation is the only true path to achievement, that only by being flawless can you succeed. To compound this, he withheld his affections when these tenets were not expressed to his satisfaction.
Secondly, you have Ursa doing the mothering -a woman who very clearly neglected Azula in favor of Zuko. We'll not get into Ursa's reasoning too much here (though it was likely she thought she could only save one of her children and so picked Zuko, the weaker of the two offspring), but the end result is that Ursa's failure to counteract Ozai's influence was a significant portion of Azula's poor upbringing.
So you've got a young girl growing up being told that being a flawless sociopath is the only way to get through life, busting her ass every day just so her sole parental figure will show her even the slightest of affections. Meanwhile, her pathetic loser of a brother fucks up every day of his life and gets showered with love by their mother. By the time she starts to realize that everything daddy fed her was a lie, it's too late; her friends/minions betray her and the world starts to collapse around her feet. When Ozai himself even screws her over, she comes to understand how completely fucked everything is and just can't reconcile what she's been told everything is with how things really are. The already weakened foundation of her mind gives way entirely.
That's why she broke.