I'm studying Master in Korean, I will graduate in March 2016. My aunt invited me and also guarantee any costs during the travel time. I just have transcript, family ties documents. So what is the necessary to prove that I will not go to US for living?
Show your school transcript, and maybe print out potential job postings that you're considering. Also show them that you speak Korean and you have decent chances to get a job in Korea (since you're applying for a visa, I'm assuming that you're not Korean, cause you'd be eligible for VWP. I'm also assuming that you live in Korea).
Just be honest. The main thing is that you need to show them that you're not going to stay in US illegally. Living South Korea is quite an advantage, since it's also a developed country (i.e. why would you stay illegally in US when you can stay legally in Korea), so I wouldn't worry much. I remember my interview in an embassy in Frankfurt when I had Azerbaijani passport was shorter than interviews of Germans. I had a German permanent residence back then, so it showed that I don't break immigration laws.