I believe @Cronax is right. She wants to feel loved and supported.
Some people when angry simply want you to listen to them vent. Others need more; the reason they're angry might be that it triggered something in them that hurts, and their hurt has turned to anger. This is harder to deal with, because you need to figure out the underlying hurt to support them. Another type really is asking if they were correct. This is not the majority, so telling someone their anger is wrong is an ill-considered move.
When people close to you ('you' meaning anyone in general) are angry, it's uncomfortable. Few people enjoy it, so many want to put a stop to it as soon as possible. So, some people offer or try to "fix" the problem.
...I ask her to calm down... getting angry won't resolve anything and she should not even think about her/him... I try to tell her that he or she might be doing it because of this or that reason.
That's trying to fix the problem, but since anger is rarely simple, this doesn't work.
I love her a lot and I genuinely care for her, that's why a lot of the time I keep telling her that she should stop getting angry because that's a bad thing and in return she starts hating me.
Imagine this scenario (totally fictional): Imagine your girlfriend was always told she was wrong as a child. Imagine that as a result, when she's told as an adult that she's wrong, it puts her right back in that place where she had no power to fight back when she was told she was wrong, always wrong, to do this, that or the other thing. It's not a pretty place to be, and she's reacting emotionally, not intellectually, whereas you're treating her like this is an intellectual discussion. It's not.
How to handle my girlfriend when she takes things wrong way?
If you want to stay in this relationship, you need to do a few things.
First, let her vent and listen. Support her where you can. Point out where she was right, point out when the other person was wrong (not the opposite!) Her anger will dissipate much more quickly if you do this. Then, after she has been calm for a (long) time, ask her why she felt so angry about x, because you want to understand her better. You might see a pattern emerge that will be very helpful.
Second, don't tell her when she's angry that she's wrong. Just don't, even if she is.
Third, read about boundaries, what they are, how to set them, how to enforce them. When two people love each other, one person's problems are also the other person's problems, until it crosses a line into co-dependency. That line is the boundary between healthy and unhealthy.
An example of an unhealthy expectation is that you should stay up all night talking with her about her issue. That's an unrealistic and ultimately unkind expectation.
Setting healthy boundaries are vital to healthy relationships. You will feel less trod upon, she will be less likely to include you in her anger.
What's an example of a healthy boundary? For starters, when you are both in a good place, talk about this:
I love you, and I will listen to you when you are upset. But I need sleep to function well the next day. Unless it's a serious emergency, we need to stop talking about your problem at a reasonable hour, and pick up the conversation the next day if needed. But I need my sleep, and you need to let me sleep.
If she agrees (and she should), then great. Next time she tries to keep you awake, remind her of the agreement. If she gets upset, you do not. Remind her that you love her, but you were serious, you need your sleep, and you'll talk about this tomorrow. If she won't let it go, get up and sleep in another room. If she follows you into that room, you know there's a bigger problem here that you cannot fix by talking.
Healthy boundaries will help her. Understanding where her anger comes from will help both of you. If you can point out in a supportive way why she might be angry, in a way she'll hear and feel understood, she will be less likely to stay - and maybe even become - angry.
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What Are Boundaries
Having a conversation to assert your boundaries
