This is a fascinating question, especially within the context of yoga, given that one of the primary aims of yoga is to know thyself. The practice of self-study, known in yogic speak as svadhyaya, is one of the core teachings of yoga’s beloved and vast tradition.
We cultivate awareness so that we can see our automatic patterns of behavior, as well as our internal stories — all the thoughts in our head that restrict us from seeing the real nature of ourselves as well as the universe. The path of the yogi is to see everything as One. And we can only do this once our perception becomes crystal clear. One of the most effective ways to clean our internal doors of perception is to continuously become more and more aware of ourselves, until we realize our true nature.
To cultivate self-awareness, we can do a lot of things. And these things can actually be quite fun! If I could choose just one way to become more self-aware, I would meditate every single day. There are so, so many ways to meditate and on some yoga websites, instructors even offer up free meditation practices that guide you into self-awareness!
(Here's the answer to a related question: What can meditation tell me about myself?)
The yoga asana practice itself also helps you become more self-aware. It allows you to slow down and tap into your body and breath. In this way, you get to know yourself, your body, your mind and even your emotions. So, do a bit of physical yoga practice daily in conjunction with a regular meditation practice.
By simply slowing down and decreasing the amount of distractions around you, you’ll naturally become more self-aware. Do this by eliminating external noise in your environment, everything from music to television. Take walks in nature as often as possible. Spend time by yourself, journal your thoughts in a yoga journal. Any contemplative practice will guide you to self-awareness. All you have to do is spend time each day doing it.
(Read on for the answer to a related question: Should I be keeping a yoga journal?)