How should an graduate student with Aspergers syndrome prepare for a teaching career?

I think what I said are largely "abstract nonsense" everyone knows.

I'd bet that's part of the issue here. When people ask a question, the general assumption is that the answer will try to maximize usefulness. When this doesn't seem to be happening, it's viewed as a very strong signal. For example, suppose I ask you "Are you having lunch with John tomorrow?" and you reply "I don't know. Maybe he'll commit suicide tonight." If there's no special risk of suicide, then this is a useless answer, since everyone knows suicide is a theoretical possibility that would disrupt the lunch plans. So it's natural to interpret this answer as suggesting you honestly fear John will kill himself. Giving a logically correct but near-vacuous answer is considered highly eccentric (if not done deliberately) or rude (if done deliberately). Either way it will upset people, because it completely throws off their ability to judge what is really meant. It can be OK as a joke under the right circumstances, but otherwise it's generally problematic.

So there's a real danger whenever you try to explain "abstract nonsense everyone knows". It can be misinterpreted in many ways. "He thinks I'm an idiot who needs to be patiently told things everyone knows", "He's making fun of me", "He's intentionally being rude", "He's gone crazy and can't stop talking about suicide and the bell curve even though they are irrelevant to my needs", "He's severely depressed and is trying to prepare me for the possibility that he might be gone by tomorrow [or is awkwardly asking for help]", etc.

Math culture can play a role here as well. It's not uncommon for mathematicians to be a bit obsessive about logical correctness in everyday life. For example, when someone asks me about my lunch plans, I have to fight to urge to reply "as far as I know" instead of "yes", since of course I don't really know for sure. If I see a bag sitting my itself in my classroom as everyone is leaving and ask the class "Did somebody leave their bag behind?", I'm sure to get an answer of "yes" from multiple students. Among mathematicians, this communication style is widely tolerated, and it can be viewed as amusing or a sign of in-group solidarity, but it can really provoke non-mathematicians.

So it's important to keep in mind that Asperger's and math culture can line up in ways that interfere with effective communication. This is usually less important in advanced classes, but it can be a big deal in introductory classes, especially with students who are not majoring in mathematics.

Dealing with this can be nontrivial, but I'm confident you are on your way to sorting it out. As I see it, the two biggest obstacles are realizing that there's an issue and recognizing that it's more than just "students can be irrational", and you've made it past both of these obstacles.

Even if there is no one mentioning this on my future teaching reference, would I be qualified for a teaching position in future once I got out of grad school?

Yes, Asperger's is far from rare in mathematics and it is in no way a disqualification for a teaching career. Overcoming initial hurdles can lead to a very effective teaching letter, since this demonstrates a serious and professional commitment to teaching.

Many universities have a teaching center or the equivalent, where people can go for feedback or assistance with their teaching. For example, someone could sit in on your class and offer advice, or video a session and go over it with you (seeing yourself teach from a student's perspective can be illuminating). It might also be possible to be paired with a long-term mentor. There can be advantages to working with someone outside the math department (who is not involved in evaluating you professionally) or within the department (who can offer better feedback on the clarity of your mathematical explanations). Either way, it's worth looking into what resources might be available, since this can be a valuable way to improve and to demonstrate commitment.

First of all, kudos to the OP for sharing his problem. Without a doubt, this the first step towards addressing the problem. Unfortunately, most of the answers do not seem to get to the "meat" of the problem. And the main question is:

Was the OP rude to the student and was the student on the right to complain about this behavior?

The answer to both questions is a big YES. When someone asks a question about a test quiz and gets an answer like "I do not want to tell you because I might commit suicide because of depression" the student (in his human imperfection), has the right to feel "scared" and "awkward". But even then, the student still made the right choice: He notified the university "authorities" (the main instructor of the course) about the problem he faced. This is what we suggest all students to do (in this SE forum) when they are dealing with a dangerous or threatening situation within the university campus. And although the danger was not real, the student felt that way, so he reported it. What would a teacher do, when one of his students told him when asked about a meeting, "I do not know if I can make the meeting tomorrow because I might be dead"? Exactly the same thing. Notify the university authorities. So first, I believe we must all agree that the student did the right call. Otherwise, the OP might not even have acknowledged the problem and how the student felt from this situation (which I am not sure if he totally gets is 100%). And BTW commenting on "I do not want people to be unhappy with me for trivial events like this in future", let me tell the OP that being "scared" by your university teacher is not a trivial event at all.

I am not a doctor and I cannot provide remote diagnoses. No one should do it especially online. But being a teacher is mostly about the students and not ourselves. We are still human, we are imperfect and we sometimes make irrational mistakes. But if something (disease, personal state, even grief) prevents us to do what is best for THEM, we should do whatever it takes (therapy, medication, personal leave) to protect them and be the best teachers we can be. Otherwise, we are not doing those young people justice.

+1 to Anonymous' answer.

In addition, you mention that you have Asperger syndrome. One of the key symptoms in Asperger - right in the first paragraph at Wikipedia - is "significant difficulties in social interaction and nonverbal communication". That is: your interaction with the student was likely "abnormal", to use the term that you used. And importantly, you probably have a hard time in understanding just what it is that caused the student distress.

What could have been "abnormal"? This is hard to say. Communication happens on many levels. Only a very small percentage (I have seen suggestions of 10%) are the actual information content. The rest is "everything else": choice of words, intonation, facial expression, body language, context, surroundings, relationships and so forth. It is very hard to explain what really goes on here. And of course your 10% information content could have been exactly on the mark, while the other 90% could have been just a bit outside what the student expected within the parameters of such a conversation. And that would be enough for him to conclude that you are "crazy". (Although of course telling this to your professor is rude.)

So, what can you do? One problem with Asperger is that you will need to consciously learn and practice communication norms that non-Aspergers soak up automatically, so they don't even need to think about them. I am not a therapist, so take my advice with a large grain of salt. But I would make sure that you stick with the actual information you need to convey (as Anonymous recommends). Given that you have problems in understanding how you come across, better to reduce the possibilities of being misunderstood. Don't do jokes, don't discuss six-sigma events like a meteorite hitting you.

In addition, I would strongly recommend that you look for help. Either talk to a therapist - student services at your university could probably help you there. Or at least look for a self-help group. Asperger's is not too rare, so you may find such a group close by. If you have a trusted (non-Asperger) friend, ask him to observe you in conversation and ask him to give you feedback. Maybe even take a video of you in an interaction so the two of you can go over it in detail.

If you work on your communication skills, I don't think you would be barred from teaching. One question is whether you actually enjoy it - many people with Asperger's don't enjoy interaction with other people.