Some characteristics that I believe make a teen realize they are maturing are, for one, definitely becoming more independent and responsible. Adulthood is full of responsibilities; In order to handle these you need to be able to accept the responsibilities, and then realize that, " Hey, my mom doesn't have to make me wake up and get ready, or force me to do things I should already know to do," they realize that they are being more responsible and adult like.
Teens also can tell when they are becoming more adult like, because they look deeper into things, and not just at what is on the surface. They become like a scientist and dissect things down so they can probe it for more information. In doing this, teens gain a deeper understanding of life. They no longer will view life as a task but more as a challenge to make you better.
To be honest I don't think anyone knows for sure the exact second they become an adult. It is a very gradual process that most of us never even realize is happening, or has happened already. Looking back now, I have noticed a very large change in the level of my maturity. While in Junior High, I never thought about the results due to my actions. Looking back I see that now I have gained the maturity to be able to look at these things, and now I always think of the results before I engage in the action. I don't know if I'm an adult or still a child. In my opinion, the change happens so slow, I don't think anyone really knows for sure until it is already over.