Experiencing ourselves as a separate individual is necessary to understand us better. We then can empower ourselves in the best way possible. What a life could be without knowing thyself? Talking with our inner self, being connected with it is a powerful tool. But as it's not popularly practiced so we don't also realize the necessity.
Self-awareness can create a chain to bring every good thing in our life. It's not like that it will help us to achieve our goal all of a sudden. But it surely can make us more able for that and also more patient for it. Another important role of Self-awareness is, it helps to gain the power to achieve the mental strength to accept everything and be happy with what we have.
Self-awareness is critical. Only observing our behavior and activities is not enough. Many other things are related to it. And it's not something we can achieve overnight. It's more like a through process that we need to include in our regular life. For me, mindfulness practice is the best way to Self-aware. Being present to every moment, feel every emotion and take note of it, observe thyself towards different feelings, practicing meditation can help a lot.
It's a very vast thing. But we can start with a little. I believe, observing how we react and respond to anything, being present towards our surroundings and how we respond to them can be the starting step. Thus we can gradually grow the practice to be in the moment. Also, seeing ourselves objectively can help a lot. Like seeing us as a person from other's end can help us realize who we are, how we can improve.